Integral calculus is a fundamental branch of mathematics that deals with the concept of accumulation and the reverse process of differentiation. Integrals are used to find the area under curves, calculate total quantities, and solve a wide range of problems in science, engineering, economics, and various other fields. While solving integrals by hand can be challenging and time-consuming, integral calculators have become indispensable tools for students, professionals, and researchers. In this comprehensive guide, we will explore integral calculators, their functionality, and how they can be used effectively at Myassignmenthelp.com
I. Understanding Integration
Before delving into integral calculators, it’s essential to grasp the fundamental concept of integration. At its core, integration involves finding the integral of a function, which represents the area under the curve of that function within a specified range. This process is often denoted by the integral symbol (∫) and is typically expressed as:
∫ f(x) dx
- ∫ signifies the integration operation.
- f(x) represents the function you want to integrate.
- dx is a differential element that signifies the variable of integration.
Integration is used for various purposes, including finding the area between curves, calculating volumes of irregular shapes, determining the accumulation of quantities over time, and solving differential equations.
II. Manual Integration vs. Integral Calculators
- Manual Integration: Manual integration involves using techniques like the power rule, substitution, integration by parts, and partial fractions to solve integrals by hand. While it’s essential to learn these techniques for a deeper understanding of calculus, they can be time-consuming and prone to human error, especially for complex functions.
- Integral Calculators: Integral calculators, also known as online or software-based calculators, automate the process of finding integrals. These tools are designed to handle a wide range of functions and provide fast and accurate results. Integral calculators offer several advantages:a. Speed and Efficiency: Integral calculators can solve integrals within seconds, even for complex functions, saving valuable time.b. Accuracy: They eliminate the possibility of human errors in the integration process, ensuring precise results.c. Accessibility: Integral calculators are readily available online and on various devices, making them accessible to anyone with an internet connection.
III. How Integral Calculators Work
Integral calculators utilize numerical techniques to approximate definite and indefinite integrals. The process generally involves the following steps:
- Input Function: Users input the function they want to integrate into the calculator. Some calculators also allow for the specification of integration limits for definite integrals.
- Algorithm Selection: The calculator selects an appropriate integration algorithm based on the complexity of the function. Common methods include the trapezoidal rule, Simpson’s rule, and adaptive quadrature.
- Calculation: The calculator numerically evaluates the integral using the chosen algorithm, breaking the function into smaller segments and summing their contributions to approximate the integral value.
- Display of Results: Finally, the calculator displays the result, which can be either an approximate numerical value or a symbolic expression for indefinite integrals.
IV. Types of Integral Calculators
Integral calculators come in various forms, catering to different user needs:
- Online Integral Calculators: These are web-based tools that can be accessed through a browser. They are convenient for quick calculations and are often free to use.
- Graphing Calculators: Many graphing calculators, such as the Texas Instruments TI-84, have integral functions built into them. Users can input functions directly or graphically and find integrals on the device.
- Software-Based Calculators: Some software programs, like Mathematica, MATLAB, and Maple, offer integral calculation capabilities. These tools are powerful and suitable for more advanced mathematical tasks.
- Mobile Apps: Integral calculator apps are available for smartphones and tablets, making it easy to perform calculations on the go.
V. Effective Use of Integral Calculators
To make the most of integral calculators, consider the following tips:
- Understand the Problem: Before using an integral calculator, ensure you understand the problem and the type of integral you need to calculate (indefinite or definite).
- Input Carefully: Be precise when inputting the function into the calculator. Errors in input can lead to incorrect results.
- Check Results: Always verify the calculator’s output, especially for complex integrals. Compare the result with your manual calculations if possible.
- Learn from the Process: While integral calculator are incredibly useful, take the opportunity to learn from the process by studying the steps and results. This can deepen your understanding of calculus.
- Select the Right Calculator: Choose the type of integral calculator that suits your needs. For basic calculations, online calculators are sufficient, while advanced tasks may require specialized software.
Integral calculators are powerful tools that have revolutionized the way we approach integration problems in mathematics and science. They provide quick and accurate solutions, making complex calculations accessible to a broader audience. Whether you are a student learning calculus, a professional in a technical field, or a researcher conducting advanced mathematical analyses, integral calculators are indispensable for solving a wide range of problems efficiently. By understanding how they work and using them effectively, you can harness the full potential of these valuable tools in your mathematical endeavors.


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Core Courses (3-Year Sequence): Algebra I1, Geometry, Algebra II ... 7th-8th Holt Middle School Math, McDougal Littell Pre-Algebra, McDougal Littell Algebra, McDougal Littell. Geometry exponents to those values, allowing for a notation for radicals in terms of . Explain why the sum or product of two rational numbers is.
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The cartisian circle
Find and save ideas about cartesian coordinates on pinterest | see more ideas about equation of plane, integers activities and plane math. Coordinates in this form are called polar coordinates so, this was a circle of radius 4 and center (-4,0) this leads us into the final topic of this section. The cartesian circle is a potential mistake in reasoning attributed to rené descartes descartes argues – for example, in the third of his meditations on first. [this is a graduate seminar paper, published in summa philosophiae (1997) --mh] the cartesian circle persists i have one further worry, namely how the author avoids. You can use absolute or relative cartesian (rectangular) coordinates to locate points when creating objects to use cartesian coordinates to specify a point, enter an. Converting equations polar & cartesian the idea in the last couple sections we’ve established convert the following equation of a circle to polar coordinates.
Nlab cartesian space skip the navigation links | home page circle, torus, annulus, moebius strip polytope, polyhedron projective space (real, complex. According to philosopher and mathematician rene descartes, the cartesian plane is formed when two perpendicular number lines intersect to form a graph of data. Sectiond2 the cartesian plane the cartesian plane • the distance and midpoint formulas • equations of circles the fixed point is the centerof the circle. This is from an old floppy disk i recently found in my attic, but it appears to be very close to how this paper appeared in the 1992 pacific philosophical quarterly. The gist of cartesian-style skeptical arguments is that some empirical proposition (eg that there are trees) cannot be known because we might be deceived (eg we. Verify that all complex numbers z satisfying the equation zz = r2, r0 a real constant, lie on a circle what is its centre and radius 5 ∗39.
I'm trying to draw a function's curve, so i need a method to convert my curve points coordinates to screen coordinates but i can't get it to work here's the method i. I am wondering what will be the equation of a triangle with vertices at (1,0), (0,-1), and (2,-1) i really appreciate your quick responses on this shah.
Cartesian definition, of or relating to descartes, his mathematical methods, or his philosophy, especially with regard to its emphasis on logical analysis and its. Polar and cartesian coordinates and how to convert between them in a hurry read the summary but please read why first: to pinpoint where we are on a map or. Online math dictionary: c a circle is the set of points that all lie within a given distance of the center of the circle this given distance is called the.
The cartisian circle
Cartesian and polar coordinate systems introduction our second coordinate measures an angular distance along the circle we need to. Cartesianism: cartesianism, the philosophical and scientific traditions derived from the writings of the french philosopher rené descartes (1596–1650.
Graphing and equations of two variables graph of [latex]x^2+y^2=100[/latex]: this is a graph of a circle with radius 10 and center at the origin example 2. The cartesian circle written by tutor steve c there are three locations for graphing a circle in the xy cartesian plane: at the origin, on the edge, and anyplace else. Students learn about an important characteristic of lines: their slopes students get an explanation of when and how these different types of slope occur. A cartesian plane (named after french mathematician rene descartes, who formalized its use in mathematics) is defined by two perpendicular number lines:the x-axis. Coordinate geometry notes full set coordinate plane formulas 1 determine the equation of the line through the center of the circle and the midpoint.
Tex - latex stack exchange is a question and answer site for users of tex, latex, context, and related typesetting systems join them it only takes a minute. Conic sections in cartesian and polar coordinates possible, though, for a to equal b (in which case, the ellipse becomes a circle), so that f = 0. Where when the mouse is above and to the right of the particle (center circle) such as : the console log displays: cartesian to polar coordinates resulting in nan 0. How to transform polar to cartesian learn more about polar, cartesian, image processing. Infallibilism and the cartesian circle michael lacewing infallibilism and certainty knowledge is more than true belief a belief can be accidentally true, true. Cartesian circle: cartesian circle, allegedly circular reasoning used by rené descartes to show that whatever he perceives “clearly and distinctly” is true.


Not often do you combine math with rock climbing and kayaking, but this past summer we did just that. The Math for America San Diego Math + Leadership Camp was a 10-day summer camp for 9th-11th grade students from the Oceanside and Vista Unified School Districts. Held at CSU San Marcos this past July, the camp provided mathematics classes in the mornings and leadership and team building activities in the afternoons with arc.
Housed in the Center for Research on Educational Equity, Assessment and Teaching Excellence (CREATE) at UC San Diego, MfA SD works closely with county school districts to develop and deliver high quality, mathematics teaching and learning programs to mathematics teachers and students. MfA SD, in conjunction with UC San Diego’s CREATE STEM Success Initiative, focused the summer work on helping students regain their enthusiasm for mathematics while enjoying some outdoor activities.
In past summers, MfA SD held mornings-only math academies. This year, the program wanted to expand to an all-day camp experience combining mathematics learning with more traditional summer camp activities.
“With a big assist from Brenda Hall, director of the North County Professional Development Federation, we were able to connect with Brad Lupien, co-founder and co-president of arc,” said Barbara Edwards, executive director of MfA SD. “Brad and his team designed the ‘plus Leadership’ part of the camp. We were delighted to integrate arc’s teamwork and leadership building program, including exciting, off-campus field trips, into the afternoon portion of the camp.”


This week the children will be innovating the model text. They will be researching and writing a persuasive text on how to save the whale.
All of the work in this unit will be based on the Michael Morpurgo book 'This Morning I met a Whale.'
In Number lessons the children's focus will be using and applying knowledge of percentages and recapping prior learning from previous years.
In shape lessons the focus will be on symmetry, rotation and translation.
We will be looking at a range of fluency, reasoning and problem solving tasks too.
|This week in IPC ..|
Our unit of learning this half term is 'Fairgrounds.'
|Parents can help their child by…|
*Recapping the passport tasks for your child's next passport.
*Encouraging them to continue to develop speed and accuracy in times tables and division facts for up to 12x12.
Home work will be scheduled each week using Google Classroom
Please check that your child knows their log in details.
Children are expected to read at home to an adult at least three times a week
Spellings are given out each week for the children to learn.
Please encourage children to log in to their TTRockstars account to complete their challenges.


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Gender Gap in Science Education
➢ February 11 was the International Day of Women and Girls in Science, established by the United Nations to promote equal access to and participation in science for women and girls.
➢ According to a 2018 fact sheet prepared by UNESCO on women in science, just 28.8% of researchers are women.
➢ UNESCO data from 2014-16 show that only around 30% of female students select STEM (science, technology, engineering and mathematics)-related fields in higher education.
➢ Female enrolment is particularly low in information technology (3%), natural science, mathematics and statistics (5%) and engineering and allied streams (8%).
➢ Various studies have found that girls excel at mathematics and science-oriented subjects in school, but boys often believe they can do better, which shapes their choices in higher studies.
➢ In 2015, an analysis of PISA scores by OECD found that the difference in maths scores between high-achieving boys and girls was the equivalent of about half a year at school.
➢ But when comparing boys and girls who reported similar levels of self-confidence and anxiety about mathematics, the gender gap in performance disappeared, when girls were more anxious, they tended to perform poorly.
➢ The NITI Aayog report said, “The problem of entry of women in science is not uniform across disciplines.


Celebrate Pi Day: Do You Know Why and How it Started
We cannot deny the significance of Formulas in our technical & scientific developments. Today whole world marked a special mention for a mathematical formula like π (Pi).
Pi Day is an annual celebration of the mathematical constant π (pi). Pi Day is observed on March 14 (3/14 in the month/day date format) since 3, 1, and 4 are the first three significant digits of π. In 2009, the United States House of Representatives supported the designation of Pi Day.
Pi Approximation Day is observed on July 22 (22/7 in the day/month date format), since the fraction 22⁄7 is a common approximation of π, which is accurate to two decimal places and dates from Archimedes.
Google celebrates it with a special doodle and designed home page.
Read More, about recent activities and news at BEYOND TEACHING.


If a 4.5 feet person (who does not know swimming) is to cross a river which at its banks is 2 feet deep, and in the middle 6 feet deep.
If he is to consider the average depth, then it would turn out to be 4 feet deep. So as the person is 4.5 feet, and if he crosses the river just because it has an average depth of 4 feet, he would drown when he reaches the middle of the river. This is the problem with the averages. It usually hides the real scenario and it is usually difficult to determine anything properly by using averages. Thus it becomes very important to use variation and targets instead of averages while doing Quantitative Project Management. One example where I can think of where average is used considerably well and across many organizations is the Customer Satisfaction Survey (CSS). With the known issue of averages, what would be the best way to determine the CSS?


Random Number Generators: what do you need one for
Random Number Generators are a vital part of all modern operating systems. Being computers a deterministic system, random numbers are used where a non-deterministic input is required. An obvious use is virtual dice rolling for gambling systems, but if you donít plan on running an online casino, what do you need one for?
Random numbers are utilized in many different areas, ranging from cryptography (in general) to source port and process ID randomization in some operating systems. So what is exactly a Random Number Generator?
Random number generators are just a source for random numbers. In their most basic form, they are functions that return a random number within a range.
(Submitted by Flavio Villanustre Thu Sep 8, 2005 )
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During the Forum of Philosophy – Reflections on China-West Cultural Exchanges
Held in Athens 7-10 August, 2013
The geographical partition of Greece both mountains and numerous islands dictated its economy, which upon the islands, due to commercial travels the trade attended a remarkable increase. Moreover this geographical partition also dictated the representation of a govermental system , which was based on the structure of the city-state. Thus by definition the city-state controlled a limited number of the population, namely around a few thousand. Either monarchical or democratical , often referred only by name , the city-state was ruled by laws and usually possessed a flourishing economy .
Around the 6th century B.C. this was the case of the Ionian coasts where the wise men ( the word philosopher was introduced in Plato’s epoch ) abolished the cosmogony for the cosmology by studying the real constitution of nature . This movement is known as the passage from the mythological interpretation of nature to the rational one. Thus the rational inquiry commenced in order to attend the truth in philosophy, whereas mathematics constituted its subset. Upon these coasts was born the concept of mathematical proof as a turning point in the history of mathematics, as well as in the history of science.
From the History of Eudemus we are informed that the concept of mathematical demonstration is attributed to Thales of Miletus ( c.624-597 B.C.), who “discovered many propositions and disclosed the underlying principles of many others to his successors “.
Pythagoras ( c.572-497 B.C ) who for political reasons abandoned his native island of Samos ,formed a philosophical school in Croton , in Southern Italy. He created a mathematical type of philosophy whose main doctrine was that ” the integer number is the substance of all things “. In reality the Pythagoreans considered that the basis of all natural phenomena was ruled by numbers , as well as the basis of the universe. Moreover Pythagoras discovered that the harmonics of music were expressed by numerical ratios.
The Platonic Academy (founded in Athens c. 385 ) constituted the highest institution , where scholars under the guidance of Plato ( 429-347 B.C.) were initiated to philosophy, as well as to mathematics . In the Republic, he discussed the type of an appropriate education in order to rule the ideal state. First of all a future monarch is required to learn: arithmetics , geometry ( plane and solid ) , astronomy and harmonics (music ) .” It has a great power of leading the mind upwards and forcong it to reason about pure numbers ,( while ) refusing to discuss collections of material things which can be seen and touched “( Plato , Republic VII 525 transl. by F. Conford . London Oxford University Press 1941) It may be stressed that for Plato , the study of geometry : drwas the soul trowards thruth”.
Euclid’s Elements of Geometry constrituted one of the main themes of the ancient Greek inheritance, which was ruled by the Aristotelian logic. It remains aa an incomparable model of pure mathematics with precise definitions , well ordered axioms , theorems and logically coherent proofs.
At the third and early second centuries B.C. Greek mathematics were dominated by Archimedes of Suracuse (c.287-212B.C.) and Apollonius of Perge ( c.250- 175 B.C.). The first one conceived and elaborated the beginnings
of the infinitesimal calculus , from the ” limit” methods of Eudoxus . He also discovered the law of the lever and its applications in order to find centers of gravity and the basic principles of hydrostatics.
Apollonius in his main treatise the Conics were developed in a corpus the important properties for these curves, which later in the hands of Kepler became thr ideal instrument to express the laws of planetery motions .
Hypatia (c.355-415), Theon’s daughter, marked the end of Greek mathematics.She taught Plato and Aristotle, considering philosophy as a way of religious misticism. She also completed her father’s commentaries on Ptolemy’s Almagest , she worked on the edition of Archimedes’ measurement of the circle , on Apollonius’ Conics and on Diophantus’ Arithmetica. Moreover she trid to preserve the ancient Greek cultural inheritance in Alexandria , where all the institutions the ,Museum .the library and the temples collapsed due to fanatisme of the forts christians .
Ancient Greek and Chinese mathematics: a few commom aspects
The initiation of the rulers to mathematics
The measurement of the hight of a pagoda in Liu Hui’s treatise and the measuremnt of the pyramid by Thales
The mysticism of numbers
Abacus ( suan phan ) and Salamis Abacus
Geometry ( mohist definitions ) – Euclid’s definitions
Theorem of Pythagoras ( Hsuan Thy )
Evaluation of π ( Han arithmetics )


Art of Problem Solving Academy Now Open and celebrates with Gaithersburg-Germantown Chamber Ribbon Cutting
Gaithersburg /Germantown, MD (August 11, 2017) – The Gaithersburg-Germantown Chamber of Commerce (GGCC) supported Art of Problem Solving Academy (AoPs) and celebrated the grand opening of their academy by conducting a “Ribbon Cutting” Ceremony for them on Friday, August 11, 2017. They are located at 9711 Washington Boulevard, Suite 150 in Gaithersburg.
Art of Problem Solving Academy offers small, highly interactive math and language arts classes for motivated learners in grades 2-12 after school and on the weekends. Fact: hundreds of thousands of the world’s strongest math students have trained with Art of Problem Solving’s textbooks since 1993 & most of the winners of major American national math competitions are AoPS alumni, and thousands of their alumni enroll in top universities each year. Learn more about them at: www.aopsacademy.org.
The GGCC is active in providing business-to-business networking opportunities throughout the year: a collective voice in local, regional, and state legislative affairs, professional development opportunities and other services that enhance the business environment.
The GGCC is located at 910 Clopper Road, Suite 205N, Gaithersburg, MD. To learn more about the GGCC visit us at www.ggchamber.org.
Find us on Facebook at: http://www.facebook.com/pages/Gaithersburg-Germantown-Chamber-of-Commerce/64339954570. Or on Twitter at: @GGCCNEWS.


Dr. Wu’s research focuses on developing and applying statistical and computational methods to address the problems in the study of clinical neurological disorders. She is especially interested in designing randomized clinical trials and observational studies, and modeling longitudinal data.
Dr. Tianxia Wu received her degree in mathematics from HuaiYing Normal University, China and her M.S. and Ph.D. degrees in statistics and quantitative genetics from Nanjing Agricultural University, China. She conducted postdoctoral research at McGill University, Canada, studying field experimental designs in agriculture. Dr. Wu was an Associate Professor in Nanjing Agricultural University. She joined NIDCR/NIH in 1999 and NINDS in 2010 as a statistician. Dr. Wu serves on the NINDS Protocol Implementation and Review Committee (PIRC) as a statistical reviewer. She provides statistical support, consultation service and statistical training to the researchers in NINDS and collaborates with NINDS investigators on clinical research.


Quadratic equations play a fundamental role in mathematics, serving as powerful tools to model and solve various real-world problems. One such equation is 4x^2 – 5x – 12 = 0, where x represents an unknown variable. This article aims to explore this specific quadratic equation by providing a comprehensive understanding of its components, methods to solve it, and its applications in practical situations.
To delve into the exploration of the equation 4x^2 – 5x – 12 = 0, it is crucial to first grasp the concept of a quadratic equation. In mathematics, a quadratic equation is a polynomial equation of degree two that typically takes the form ax^2 + bx + c = 0, where ‘a’, ‘b’, and ‘c’ are constants. The highest power of the variable ‘x’ in such equations is squared (raised to the power of two), hence giving rise to their name.
By analyzing each component of the given equation – 4x^2 – 5x – 12 = 0 – we can gain insight into its behavior and characteristics. The coefficient ‘4’ attached to x^2 determines how steep or shallow the graph will be when graphed on Cartesian coordinates. The coefficient ‘-5’ represents the linear term multiplied by x, while ‘-12’ accounts for the constant term present in the equation. Understanding these elements allows us to interpret both geometrically and algebraically what this particular quadratic equation signifies and how it can be solved effectively.
Through exploring these facets in detail, we aim to equip readers with a deeper understanding of this specific equation’s implications and broaden their overall knowledge regarding quadratic equations as a whole.
Definition of a Quadratic Equation
The definition of a quadratic equation is an equation that can be written in the form ax^2 + bx + c = 0, where a, b, and c are constants and a ≠ 0.
Quadratic equations are characterized by their degree of 2, which means the highest power of x in the equation is 2.
The coefficient a represents the leading term’s coefficient, while b represents the linear term’s coefficient and c represents the constant term.
Quadratic equations have several important properties that make them useful in various fields.
One common application of quadratic equations is in physics, particularly when studying motion. The equation can be used to model projectile motion or describe how objects move under the influence of forces like gravity.
Additionally, quadratic equations are often used in engineering to solve problems related to optimization or design. For example, engineers may use quadratic equations to determine the dimensions of structures or find optimal solutions for various systems.
Moreover, understanding quadratic equations allows for analyzing functions and their behavior. Quadratic functions have unique characteristics such as vertex form or standard form that help identify key features like turning points or symmetry axis. This knowledge aids in graphing functions accurately and interpreting their graphical representations.
Overall, grasp on quadratic equations provides valuable tools for solving real-life problems across different disciplines. By recognizing their common applications and properties, individuals can harness this mathematical concept to explore new possibilities and unlock potential solutions with precision and logical reasoning , ultimately leading to more efficient and effective problem-solving strategies.
Understanding the Components of the Equation
Examining the factors of 4x^2 – 5x – 12 = 0 allows for a deeper understanding of the equation’s components.
In this quadratic equation, the coefficient of x^2 is 4, which represents the degree or power of the variable x. The coefficient gives us information about how steep or flat the graph of the equation will be. In this case, since it is positive, we know that the parabola opens upwards.
Furthermore, there are two other coefficients in this equation: -5 and -12. The coefficient of x (-5) represents the linear term, which determines how much the graph shifts horizontally. A negative value implies a shift to the right on the x-axis when compared to a positive value that would result in a shift to the left.
Lastly, we have a constant term (-12), also known as an absolute term or free term. This constant affects where on the y-axis our graph intersects.
Understanding these components can help us apply quadratic equations in various fields such as physics. Quadratic equations are commonly used to analyze motion and calculate important parameters like velocity and acceleration. By manipulating and solving these equations, physicists can determine trajectories of objects in motion or predict their future positions accurately.
Moreover, quadratic equations play a crucial role in connecting algebra with geometry through conic sections. Conic sections are curves obtained by intersecting cones with planes at different angles and distances from their vertexes. Quadratic equations help describe these curves mathematically and allow us to understand their properties better. For instance, when studying ellipses or parabolas using conic sections theory, we often encounter quadratic equations that provide insights into their shape and orientation.
Overall, exploring the components of quadratic equations like 4x^2 – 5x – 12 = 0 enables us to grasp their significance in both practical applications such as physics and theoretical connections like conic sections. By understanding the role of each term, we can manipulate these equations to solve real-world problems and gain a deeper insight into the underlying mathematical concepts.
Graphical Representation of Quadratic Equations
This discussion will focus on the graphical representation of quadratic equations and its key points, including parabolic curves, axis of symmetry, and vertex.
A parabolic curve is a U-shaped curve that represents the graph of a quadratic equation.
The axis of symmetry is a vertical line that divides the parabola into two symmetrical halves, and it passes through the vertex.
The vertex is the lowest or highest point on the curve depending on whether it opens upwards or downwards.
Understanding these key points helps in visualizing and analyzing quadratic equations graphically.
Exploring the equation 4x ^ 2 – 5x – 12 = 0 leads to an appreciation of the beauty and elegance of parabolic curves. Parabolic trajectories, which are described by quadratic functions, have long fascinated mathematicians and scientists alike.
Here are four key aspects that highlight the significance of parabolic curves:
- Symmetry: One of the most captivating features of parabolic curves is their symmetry. A parabola is a perfectly symmetrical curve, with a line called the axis of symmetry dividing it into two equal halves. This symmetry is evident in the graph of a quadratic equation, where both sides mirror each other around this axis.
- Focus and Directrix: Another fascinating property of parabolas is their relationship with a point called the focus and a straight line called the directrix. Every point on a parabola is equidistant from both the focus and directrix. This unique characteristic allows for various applications in optics, such as reflecting telescopes or headlights.
- Maximum or Minimum Point: The vertex of a parabola represents either its highest point (maximum) or lowest point (minimum). This vertex holds important information about real-world scenarios modeled by quadratic equations, such as finding maximum profit or minimum cost.
- Wide Range of Applications: Parabolic curves find applications in numerous fields beyond mathematics itself. They can be used to model projectile motion, predict paths taken by planets around the sun, design bridges or buildings with arches for structural stability, analyze population growth patterns in biology, optimize manufacturing processes through optimization models—the possibilities are truly endless.
By understanding these aspects and exploring equations like 4x^2 – 5x – 12 = 0 that give rise to beautiful parabolic curves, we gain insights into not just mathematical concepts but also their practical relevance in diverse areas ranging from physics to engineering to finance.
Axis of Symmetry
The axis of symmetry is a significant element of parabolic curves, dividing the curve into two equal halves and showcasing its inherent symmetry.
In the context of quadratic functions, such as the equation 4x^2 – 5x – 12 = 0, the axis of symmetry plays a crucial role in understanding and analyzing the behavior of the graph.
The axis of symmetry is essentially a vertical line that passes through the vertex of a parabola, which is the highest or lowest point on the curve.
It can be determined using a formula that involves finding the x-coordinate of the vertex.
In this particular equation, 4x^2 – 5x – 12 = 0, we can find the axis of symmetry by first identifying its coefficients.
The coefficient of x^2 is ‘a’, which in this case is 4.
To determine the x-coordinate of the vertex, we use the formula: x = -b / (2a), where b represents the coefficient of x (-5 in this case).
By substituting these values into our formula, we find that x = -(-5) / (2*4) = 5/8.
Therefore, our axis of symmetry for this parabolic curve lies at x = 5/8.
Understanding and recognizing this axis helps us analyze various aspects related to quadratic functions.
For instance, if we have additional points on either side of this axis with equal distances from it (such as (-1,-3) and (3,-3)), we can determine their corresponding y-values without actually plotting them on a graph due to symmetric properties.
Additionally, knowing about this line allows us to make predictions about other characteristics like maximum or minimum values and intervals where our function may be increasing or decreasing rapidly.
Overall, comprehending and utilizing knowledge about axes of symmetry enhances our understanding and analysis capabilities when dealing with quadratic functions such as those presented in equations like 4x^2 – 5x – 12 = 0.
The vertex of a parabolic curve is the highest or lowest point on the graph and plays a crucial role in understanding and analyzing its behavior. It is often represented as an ordered pair (h, k), where ‘h’ represents the x-coordinate and ‘k’ represents the y-coordinate of the vertex. To find the vertex of a quadratic equation in standard form, such as 4x^2 – 5x – 12 = 0, we can use the formula h = -b/2a to determine the x-coordinate of the vertex. In this equation, ‘a’ represents the coefficient of x^2 term (-5), and ‘b’ represents the coefficient of x term (-5). By substituting these values into the formula, we get h = -(-5)/2(4) = 5/8.
To find the corresponding y-coordinate, we can substitute this value back into our original equation. Plugging h=5/8 into 4x^2 – 5x – 12 = 0 gives us k = f(5/8) = 4(5/8)^2 – 5(5/8) -12. Simplifying this expression yields k = -(49/16). Therefore, we have determined that the vertex of this quadratic equation is located at (h,k)=(5/8,-49/16). The significance of analyzing this vertex lies in its ability to provide valuable information about various aspects of a parabolic curve’s behavior. For instance, by examining whether it is located at a high or low point on the graph, we can determine if it opens upwards or downwards. Additionally, by studying its position relative to other points on the graph, we can infer important details about symmetry and intercepts. This analysis allows us to gain deeper insights into how changes in coefficients affect properties like maximum or minimum values.
Through this mathematical analysis, we can appreciate the significance of finding and analyzing the vertex. It not only provides essential information about the shape and behavior of a parabolic curve but also aids in understanding its symmetrical properties, intercepts, and extreme values. By studying the vertex, mathematicians can uncover patterns and make predictions that contribute to a more comprehensive understanding of quadratic equations and their applications in various fields such as physics, engineering, economics, and computer science.
Methods to Solve Quadratic Equations
There are several methods available to solve quadratic equations. These methods include factoring, the quadratic formula, completing the square, and the graphical method.
Factoring involves identifying two numbers that multiply to give the constant term in the equation and sum to give the coefficient of the linear term.
The quadratic formula is a general formula that can be used to find the solutions of any quadratic equation by substituting the coefficients into it.
Completing the square involves manipulating the equation algebraically to rewrite it in a perfect square form, allowing for easy solution.
Finally, the graphical method involves plotting the graph of the quadratic equation and determining its x-intercepts or points where it intersects with the x-axis.
Factoring the equation 4x^2 – 5x – 12 = 0 involves finding two binomials that, when multiplied together, result in the original quadratic expression. This method is based on the fact that any quadratic equation can be written as a product of two binomials.
The main goal of factoring is to simplify and break down complex equations into smaller, more manageable parts. To factor the given equation, we need to consider different factoring techniques. One common approach is to use the AC method, where we multiply the coefficient of x^2 (in this case, 4) by the constant term (in this case, -12).
We then look for two numbers whose product is equal to this value (-48) and whose sum is equal to the coefficient of x (-5). By trying different combinations, we can determine that -8 and +6 satisfy these conditions. Therefore, we can rewrite the equation as (4x + 6)(x – 8) = 0.
It’s important to note that factoring can sometimes be challenging due to various factors such as complex coefficients or non-integer solutions. Common mistakes in factoring include overlooking negative signs or incorrectly applying distributive properties during simplification. It’s crucial to carefully analyze each term and ensure correct application of algebraic principles throughout the process. By mastering different factoring techniques and being aware of potential pitfalls, one can effectively solve quadratic equations using this method.
An intriguing statistic regarding the quadratic formula is that it can be used to solve any quadratic equation, regardless of its complexity or non-integer solutions.
The quadratic formula, derived by completing the square in a general quadratic equation, provides an elegant solution to equations of the form ax^2 + bx + c = 0, where a, b, and c are constants.
Deriving the quadratic formula involves rearranging terms and isolating x in order to find its value(s). By applying this formula, one can obtain both real and complex roots of a quadratic equation.
The applications of the quadratic formula extend beyond mathematics into various fields like engineering. In engineering, there are numerous situations where solving for unknown quantities or finding critical points requires dealing with equations involving squares or quadratics.
For instance, when designing bridges or buildings, engineers often need to calculate the maximum height that a structure can reach before collapsing under its own weight. This problem can be modeled using a quadratic equation and solved using the quadratic formula.
Additionally, in electrical engineering, the behavior and stability analysis of circuits often involve solving second-order differential equations which can be transformed into equivalent forms solvable with the help of the quadratic formula.
The versatility and power of the quadratic formula make it an invaluable tool not only in mathematics but also in many other disciplines such as physics and economics. Its ability to provide accurate and precise solutions has made it a fundamental concept for problem-solving across different fields.
As we explore further into this topic, we will uncover more fascinating applications of this simple yet powerful mathematical tool.
Completing the Square
A common method used to manipulate quadratic equations into a more manageable form is called completing the square. This method involves rearranging the terms of the equation to create a perfect square trinomial, which can then be easily factored or solved.
Completing the square is particularly useful when dealing with quadratic equations that do not factor easily or have complex coefficients.
One application of completing the square is in solving quadratic equations to find their roots. By manipulating the equation through completing the square, we can isolate the variable and solve for its values. This technique is especially helpful when trying to find precise solutions, as it allows us to work with simpler forms of the equation. Completing the square also has applications in geometry and physics, where it can be used to derive formulas for finding areas, volumes, or other measurements.
Like any mathematical method, completing the square has both pros and cons. One advantage of this technique is that it guarantees an exact solution for any quadratic equation. This can be important in certain situations where approximate answers are not acceptable. Additionally, completing the square provides a systematic approach that can be applied to any quadratic equation, regardless of its complexity or structure.
However, one drawback of completing the square is that it can be time-consuming and tedious compared to other methods such as factoring or using the quadratic formula. The process requires careful manipulation of terms and may involve multiple steps before reaching a simplified form. Another limitation is that completing the square may not always yield real solutions for every quadratic equation, especially if complex numbers are involved.
Completing the square is a valuable technique used to transform quadratic equations into more manageable forms by creating perfect squares trinomials. It has various applications in mathematics and science fields and offers an accurate solution for any given quadratic equation. However, it may require more effort compared to alternative methods and may not always provide real solutions depending on specific circumstances, such as when the discriminant of the quadratic equation is negative. In such cases, the quadratic equation does not have real solutions and the perfect square trinomial method may not be applicable.
The graphical method unveils the hidden landscape of a quadratic equation, allowing us to visually explore its behavior and uncover the relationship between its variables.
By plotting the equation on a graph, we can analyze the roots of quadratic equations and gain insights into their nature. In the case of the equation 4x^2 – 5x – 12 = 0, we can plot it as a parabolic curve on a Cartesian plane. The x-axis represents the values of x, while the y-axis represents the corresponding values of y.
To find the roots of this quadratic equation graphically, we look for points where the curve intersects with the x-axis. These points represent solutions to our equation. By examining the shape and position of our plotted curve, we can determine if there are one or two real roots or if there are no real roots at all.
Additionally, comparing graphical and algebraic methods for solving quadratic equations allows us to appreciate their similarities and differences. While algebraic methods involve manipulating equations symbolically using formulas such as completing the square or applying quadratic formulae, graphical methods focus on visual representations that provide an intuitive understanding of how different values interact within an equation.
Analyzing quadratic equations using graphical methods brings forth a deeper understanding by visually representing their behavior and exploring their relationship between variables. By plotting these equations on graphs, we can identify solutions more easily by locating where they intersect with axes.
Comparing graphical and algebraic methods further enhances our understanding of these mathematical concepts by highlighting both their similarities and differences in problem-solving approaches.
Solving the Equation Step-by-Step
To solve the equation 4x^2 – 5x – 12 = 0, we will now proceed step-by-step.
The first step in solving quadratic equations is to check if the equation can be factored. In this case, it is not immediately apparent how to factorize the equation, so we will use the quadratic formula.
The quadratic formula states that for any quadratic equation of the form ax^2 + bx + c = 0, the solutions for x can be found using the formula: x = (-b ± √(b^2 – 4ac))/(2a).
Applying this formula to our given equation, with a=4, b=-5, and c=-12, we get x = (5 ± √((-5)^2 – 4 * 4 * (-12)))/(2*4). Simplifying further gives us x = (5 ± √(25 +192))/8. This simplifies to x = (5 ± √217)/8.
Now that we have obtained two possible values for x using the quadratic formula, we can conclude that there are two solutions to our original equation: x = (5 + √217)/8 and x = (5 – √217)/8. These are exact solutions and cannot be simplified further without resorting to numerical approximation methods.
By solving this quadratic equation step-by-step using the quadratic formula, we have successfully determined its solutions.
Quadratic equations find applications in various fields such as physics, engineering, economics, and computer science. They are used to model real-world scenarios involving motion or growth/decay processes. The ability to solve these equations enables us to analyze and make predictions based on mathematical models. By understanding and applying techniques like factoring or using formulas such as the quadratic formula, we gain valuable tools in problem-solving across disciplines.
Understanding the Nature of the Solutions
Examining the characteristics of the solutions to quadratic equations provides valuable insights into their behavior and implications within various disciplines.
When solving a quadratic equation of the form ax^2 + bx + c = 0, one can determine the nature of its roots by analyzing its discriminant. The discriminant, denoted as Δ, is calculated as b^2 – 4ac. This value serves as an indicator for understanding whether the equation has real or complex solutions.
By examining the discriminant, we can draw conclusions about the nature of roots in this specific equation. In particular, if Δ > 0, then there are two distinct real roots. This implies that the quadratic equation intersects the x-axis at two different points.
On the other hand, if Δ = 0, then there is only one real root with multiplicity 2. This means that the graph of the quadratic equation touches but does not cross the x-axis at a single point.
Lastly, if Δ < 0, then there are no real roots and instead two complex conjugate roots exist. Complex conjugate roots come in pairs and cannot be represented on a traditional Cartesian coordinate system since they involve imaginary numbers.
The analysis of a quadratic equation’s discriminant offers valuable insights into how it behaves and helps us understand its implications across various fields such as physics, engineering, economics, and computer science. The nature of roots determines critical information regarding quantities like time to reach equilibrium in chemical reactions or stability conditions in mechanical systems.
By utilizing discriminant analysis techniques when studying quadratic equations in these disciplines, researchers gain accurate predictions about their behavior without resorting to trial-and-error methods or approximations.
Incorporating discriminant analysis enhances our understanding of quadratic equations beyond mere computation; it provides us with essential knowledge about their characteristics and applications across numerous scientific domains while satisfying our subconscious desire for freedom from uncertainty through precise calculations and logical deductions.
Real-World Applications of Quadratic Equations
In the previous subtopic, we explored the nature of solutions to quadratic equations and gained an understanding of their two possible forms: real and imaginary.
Moving on to the current subtopic, we will delve into the practical applications of quadratic equations in various real-world scenarios.
Quadratic equations find extensive use in physics and economics, contributing to a deeper understanding of these disciplines. In physics, quadratic equations are employed to model motion, especially when it involves acceleration or deceleration. For instance, when analyzing the trajectory of a projectile launched into the air or studying objects moving under gravitational forces, quadratic equations become invaluable tools for calculating key parameters such as time of flight, maximum height attained, or range covered. By employing mathematical models based on quadratics, physicists can accurately predict motion patterns and determine optimal conditions for achieving desired outcomes in fields like astrophysics or ballistics.
Similarly, economics relies on quadratic equations to study various aspects of supply and demand curves. Quadratic models help economists analyze how changes in price impact consumer behavior by examining factors such as market equilibrium points and elasticities. These models also aid in determining production quantities that maximize profit while considering cost functions with both fixed and variable components. By utilizing quadratic equations within economic frameworks, policymakers can make informed decisions regarding taxation policies or pricing strategies that optimize productivity while minimizing social costs.
Overall, the applications of quadratic equations extend beyond theoretical mathematics into practical domains such as physics and economics. Their ability to accurately represent complex phenomena makes them indispensable tools for predicting outcomes and optimizing decision-making processes. Whether it is calculating projectiles’ trajectories or analyzing market dynamics, mastering quadratic equation techniques proves beneficial across various fields where precise modeling is essential.
Further Explorations in Quadratic Equations
Further investigations into quadratic equations reveal their applications in a wide range of practical scenarios, illuminating their significance and relevance in fields such as physics and economics.
In the realm of physics, quadratic equations find extensive use in analyzing motion and predicting trajectories. For example, when studying projectile motion, the path followed by an object launched into the air can be described by a quadratic equation. By considering factors such as the initial velocity, angle of launch, and gravitational pull, physicists can accurately determine the object’s height at any given time or its maximum height reached. Additionally, quadratic equations are utilized to calculate the time taken for an object to reach its peak height or hit the ground again.
Geometry is another area where quadratic equations play a crucial role. Quadratic equations allow mathematicians to explore various geometric properties and relationships between shapes. One application lies in determining the coordinates of points that intersect two curves or lines. This information aids in understanding intersections between different geometrical objects like circles or parabolas. Moreover, quadratic functions help measure important geometric properties such as areas and volumes. For instance, calculating the area of irregularly shaped regions often involves forming quadratic equations based on known measurements like side lengths or angles.
Further exploration into quadratic equations reveals their diverse applications across multiple disciplines such as physics and geometry. These equations enable physicists to analyze motion and predict trajectories accurately while allowing mathematicians to study geometric properties and relationships between shapes effectively. The significance of quadratic equations extends beyond theoretical mathematics; they provide practical tools for solving real-world problems in various fields of study.
Importance of Quadratic Equations in Mathematics and Beyond
Further explorations in quadratic equations have revealed their significance in various fields of mathematics and beyond.
Quadratic equations serve as a fundamental tool in numerous mathematical applications, including finding the maximum or minimum values of functions, solving optimization problems, and analyzing the behavior of systems exhibiting parabolic patterns.
Their importance extends beyond the realm of mathematics, as they find extensive use in physics and engineering.
In physics, quadratic equations are utilized to model various physical phenomena. For example, when studying projectile motion, quadratic equations are employed to describe the motion of objects under the influence of gravity. By considering factors such as initial velocity and angle of projection, one can determine crucial parameters like time of flight and range.
Additionally, quadratic equations are essential in engineering disciplines such as civil engineering and electrical engineering. They play a key role in analyzing structural stability, determining stress distribution in materials, designing electrical circuits with resonant frequencies, and solving problems related to fluid flow.
The applications of quadratic equations in physics and engineering highlight their practical significance beyond theoretical mathematics. They provide powerful tools for modeling real-world situations accurately and making informed decisions based on mathematical analysis.
By understanding the principles underlying these equations, researchers and professionals can effectively solve complex problems across various domains. Thus, recognizing the importance of quadratic equations allows us to appreciate their versatility and applicability in diverse fields where precise mathematical modeling is essential for advancements and innovations.
Frequently Asked Questions
How can I use the quadratic formula to solve the equation 4x ^ 2 – 5x – 12 = 0?
The quadratic formula is a powerful tool used to solve quadratic equations. It provides a direct method for finding the roots of an equation by using the coefficients of the equation. While there are alternative methods like factoring and completing the square, the quadratic formula offers a universal approach that works for all quadratic equations.
What are the different methods to factorize the quadratic equation 4x ^ 2 – 5x – 12 = 0?
Factoring techniques are used to break down a quadratic equation into its linear factors. In the case of 4x^2 – 5x – 12 = 0, one can use methods like grouping or decomposition. Another method is completing the square to transform the quadratic equation into a perfect square trinomial.
Can you explain the concept of discriminant and how it helps in determining the nature of the solutions for the equation 4x ^ 2 – 5x – 12 = 0?
The discriminant plays a crucial role in determining the nature of solutions in quadratic equations. By using the quadratic formula, which incorporates the discriminant, one can determine whether the equation has real solutions, imaginary solutions, or no solutions at all.
Are there any special cases or patterns in quadratic equations that can make solving them easier?
There are several common mistakes made when solving quadratic equations, such as forgetting to distribute or incorrectly applying the square root property. Tips and tricks for simplifying complex quadratic equations include factoring, completing the square, and using the quadratic formula. By employing these techniques correctly, solving quadratic equations can become easier and more efficient.
Can you provide some real-world examples or applications where quadratic equations are used?
Quadratic equations find practical applications in engineering and physics, such as determining the path of a projectile. In finance and economics, they are used to model profit and cost functions, optimizing production levels or finding break-even points.
In conclusion, the exploration of the equation 4x^2 – 5x – 12 = 0 has shed light on the intricacies and significance of quadratic equations. Through a precise and logical examination, we have defined what a quadratic equation is and dissected its components.
The graphical representation of such equations has provided us with a visual understanding of their behavior. Moreover, we have delved into various methods to solve quadratic equations, including factoring, completing the square, and using the quadratic formula. By following these step-by-step procedures, we have successfully solved the given equation. This process has allowed us to gain insight into the nature of solutions; in this case, we discovered two distinct real solutions.
Furthermore, our exploration extends beyond theoretical knowledge as we explore real-world applications of quadratic equations. From physics to finance and engineering to architecture, these equations play a crucial role in solving practical problems. Their ability to model parabolic shapes makes them invaluable tools in many fields.
As our journey through quadratic equations comes to an end, it is evident that these mathematical concepts are not confined solely to textbooks or classrooms; they permeate numerous aspects of our lives. Their importance cannot be understated as they provide us with tools for problem-solving and offer insights into complex phenomena.
By exploring further avenues within this fascinating realm of mathematics, one can unravel even more profound connections between theory and practice.


Numerology – the study of numbers and their connection to life patterns – fascinated me for many years. I came across a book, Numerology and the Divine Triangle which explained the Pythagorean theory of numbers. Like a psychic detective, I set out to uncover the mysteries of life, hoping, as we all do, to gain enlightenment and ultimately a modicum of control.
Of course, that is an illusion. Still, my awareness of numbers and patterns has never really subsided. This week’s challenge, for example, is #44 – a master number – supposedly noteworthy.
In numerology, 0 – 9 are the important numbers, and all other numbers can be reduced within this range. For example, to find your life path number you would add together birth month, date, and year like so:
(July) = 7
(25) = 2 +5 = 7
(1958) = 1+9+5+8 = 23 or 2+3 = 5
Total 7 +7 +5 = 19 = 1+9 = 10 = 1
So my life path number is 1.
(Token Rock has a life path calculator if my illustration is too complicated.)
Whether Numerology interests you or not, think about the numbers that have reappeared in your life. For example, throughout my marriage to my former husband, every house we lived in (and there were many) the house numbers added up to 3. Coincidence?
As always, I look forward to your answers.
To participate, create a post related to the topic of the week. Link back to this page or leave a link in the comments below. Tag your post VJWC, and remember to visit the contributions of others.


Schaefer Organic Farms purchased a new tractor at a cost of $80,000. Annual operating cash inflows are expected to be $30,000 each year for four years. At the end of the tractor’s useful life, the salvage value of the tractor is expected to be $5,000.
What is the net present value if the cost of capital is 12 percent? Use the time value of money charts for your calculations. (Ignore income taxes.) Round your answer to the nearest whole number.
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Outreach events and activities
Working closely with schools and colleges and industry, we offer inspiring and exciting events throughout the year to enhance and complement your curriculum. Our activities span engineering, design and mathematical subjects with the aim to inspire future graduates and develop their career paths.
Children as engineers
Paired Peer Mentors in Primary Schools
This project ran in 2014 as an innovative new way to enrich undergraduate education as well as giving primary school children the experience of being an engineer. Supported by the Engineering Professors’ Council, the project was organised by the Science Communication Unit and the Department of Education at UWE.
Undergraduate student engineers worked closely with student teachers to design and create exciting engineering activities for primary schools across Bristol. The paired peer mentors were inspired by the EU ENGINEER project materials to help children experience being engineers, while also improving their own public engagement skills. The model is currently being evaluated to see if it can be rolled out nationally.
Higher Education STEM days
The Department runs between five and six STEM (Science, Technology, Engineering, and Mathematics) school activity days in May and June each year, together with colleagues from Science. STEM days take place in schools and aim to reach the entire Year 8 cohort of that school.
Aims for the STEM activity day
The aims of the STEM activity days are to:
- Inspire pupils with respect to STEM subjects
- Encourage pupils to aspire to university to study STEM subjects
- Promote future careers in science, engineering and maths
- Connect with and enhance the school STEM curriculum
Depending on numbers, either two or three sessions will run, each lasting 100 minutes. UWE Bristol student ambassadors will demonstrate the equipment and run through the concepts connected with their stand to a group of 10-15 pupils every 15 minutes. This is repeated six times during each session.
Feedback from attendees
Pupils obviously enjoy it as much as we do, as seen from comments from the pupils at Redland Green school and St Bedes:
"We went around the school hall and experienced a range of wonderful workshops. I enjoyed every workshop there. Not one workshop outclassed another, so it was well laid out and explained. At every workshop I learnt something new so that was great, and every teacher at every station made sense and was good.”
"I really liked that day. I learnt a lot about astrophysics, bubbles and other stuff. I thought it was very educational and a lot of fun. It was better than lessons. I thought the bubbles one was great fun when they made a big bubble around you and the gyroscope when you have a bike wheel and you spin it fast and it makes you spin on the chair. I’d like to have another day like that.”
"I learnt that maths and science can be fun!"
"Making Aerofoils fly was great, really interesting."
Host your own STEM activity day
If you are a school interested in hosting a STEM activity day, contact Alison Hooper.
Engineering Design and Maths (EDM) event day
The EDM event day is an annual event, usually held in the third week in June. Year 12 A Level Maths pupils are invited to come to UWE for the day to:
- experience some taster sessions in mathematics and engineering
- hear from recent graduates about careers which use mathematics and engineering
- listen to a plenary talk from an inspiring speaker on applications of mathematics. Recent speakers have included Simon Singh, Kate Bellingham and Paul Shephard.
Schools interested in sending a group of up to 15 pupils to the EDM event day should contact firstname.lastname@example.org.
The Department hosts a regional round of the IET's (Institution of Engineering and Technology) Faraday Challenge annually. The IET Faraday Challenge Days are held at schools and universities around the UK and have proved to be enormously popular with teachers and students alike.
The IET is one of the world’s leading professional societies for the engineering and technology community. As a charity, they are committed to the advancement of science, engineering and technology and to encouraging young people to study the STEM subjects and to consider careers within the engineering and technology sectors.
The Faraday Challenge Days give students the opportunity to research, design and make solutions to genuinely tough engineering problems. Each IET Faraday Challenge Day involves six school teams, each made up of six 12 -13 year olds, ideally two students from Science, two students from Design and Technology and two students from Mathematics.
The winners of each event are awarded a prize for each team member and a trophy for their school. The top three teams from across the UK get an all-expenses paid trip to the National Final to compete for a cash prize of up to £1,000 for their school.
The winner of the 2013 round at UWE, The Cotswold School, came fifth in the national competition in 2013.
The Department of Engineering Design and Mathematics runs a residential summer school in partnership with Headstart, part of the Engineering Development Trust and a programme endorsed by the Royal Academy of Engineering. The course provides a project-based learning experience around BLOODHOUND, the amazing UK-led engineering initiative which aims to achieve 1000mph land speed record.
Working in teams, the students spend the week learning about the BLOODHOUND project and building their own scale model ‘high speed’ racing vehicle. By exploring the design and structure of the supersonic car and the challenges facing the BLOODHOUND team, the students get a great introduction to a range of engineering disciplines, their applications and what it would be like to study them at university.
During the week the students have the opportunity to meet with local employers, talk to recent graduates and explore a range of opportunities available to them on completion of Year 12. The week ends with a race-off with time and endurance trials. The winning team is awarded a trophy and their individual names will feature on the fin of the actual BLOODHOUND car!
The course has been running since 2011 and receives excellent feedback from those students who have attended. In 2013, more than 90% of the students said they would study a STEM subject after attending the course.
For more information about upcoming Headstart courses at UWE, contact email@example.com.
The Wessex Mathematics Masterclasses Committee (renamed from Bath and Bristol Mathematics Masterclasses) organise Mathematics Masterclasses at Bath and Bristol. The Bath series is held in the spring term whilst the Bristol series is split over both the autumn and spring term, with four classes held before Christmas and two after. The Masterclasses are run in association with the Royal Institution.
About the Bristol Mathematics Masterclasses
The Bristol series is funded by UWE and the University of Bath and has been held at UWE since 1996.
The Mathematics Masterclasses are aimed at talented and enthusiastic young mathematicians in year 8 (12-13 years). The classes are designed to show not only the many ways mathematics can be used and applied but also that the subject can be fun and interesting. Pupils will be exposed to areas of mathematics which will not necessarily be covered at school.
What happens in a typical masterclass?
In a typical masterclass, the lecturer would start by talking to the pupils for up to half an hour setting out a problem, the pupils then move to smaller tutorial classes, supervised by a schoolteacher to work on a problem. They would come back to the lecture theatre half-way through to extend the problem/start on something new and return to their tutorial rooms. Finally, the lecturer would sum up the masterclass and give the pupils a further worksheet which they can take home to do in their own time or to show to their teacher at school. Refreshments are provided.
How to take part in the masterclass
Attendance at the classes is by invitation only. In early October, local schools are invited to nominate 4 - 6 pupils from this age group to attend. We typically have 90 - 120 pupils attending per series.
Masterclasses are held on Saturday mornings from 10:00 - 12:30. Each series comprises six masterclasses led by lecturers at the Universities of the West of England, Bath and Bristol and by local schoolteachers
Contact firstname.lastname@example.org for further information.
Fun Maths Roadshows
Fun Maths Roadshows are offered to schools and colleges throughout the region and are delivered on the school/college site. Roadshows are typically run for groups of 30 to 70 pupils from Year 8 to Year 11 and sessions last up to 1.5 hours. If feasible, it is possible to run two one-hour long sessions, one after the other.
The Roadshows use mathematical activities that although graded according to age, are intended to be 'all-ability' activities that are challenging yet enjoyable. The range of interactive puzzles incorporates a broad range of mathematical ideas. Pupils are encouraged to work in pairs in an informal environment, with the emphasis on success. Participants receive a certificate of achievement.
How to take part in the roadshows
Roadshows generally take place in the autumn term and dates are communicated via e-mail. To be added to the circulation list, please email Dr Emily Walsh at Emily3.Walsh@uwe.ac.uk. Please note that roadshows are very popular and early sign up is advised.
Women into Engineering
Engineering is an exciting and well-paid career, with high graduate employability. Engineering graduates from UWE have gone on to work for companies such as Rolls Royce, Airbus, Dyson, Jaguar-Landrover, Babcock and BAe Systems. The skills needed for engineering obviously include technical skills and a strong mathematical background, but they also include skills such as communication and team-work as well as creativity and problem-solving.
Currently, there are many fewer female engineers than male engineers in the UK. We do not believe that women are any less capable than men of becoming engineers, and so we offer opportunities for girls to find out more about how their skills could be applied to engineering and to excite them about the possible careers open to engineers.
Women into Engineering sessions involve meeting current female students and recent graduates as well as hands-on activities to show the range of types of problems engineers can solve.
Contact email@example.com if you are interested in attending a session.
This year we ran a Women in Engineering Summer School to give Year 12 girls the opportunity to explore engineering in an open, friendly and interactive environment.


May 6, 2020
Dr. M. Sundarakannan – Assistant Professor
Dr. M. Sundarakannan is working as an Assistant Professor, Department of Mathematics since July, 2011. He has seven years of teaching experience and four years of research experience (as on Dec 2017). He received his B.Sc., and M.Sc., in Mathematics from Thiagarajar College, Madurai affiliated to Madurai Kamaraj University; Post Graduate Diploma in Operations Research (PGDOR) from Pondicherry University and M.Phil, (Mathematics) from Madurai Kamaraj University. He qualified GATE – 2007 Mathematics examination. He worked as a Senior Research fellow in the “National Center for Advanced Research in Discrete Mathematics (n-CARDMATH)”, sponsored by the Department of Science and Technology (DST), New Delhi affiliated to Kalasalingam University, Krishnankoil on the project entitled “Domination Related Functions in Graphs”.
He has awarded his Ph.D. thesis entitled “Equivalence Domination in Graphs” and has over seven research papers accepted in international refereed journals.
Combinatorics, Discrete Mathematics and Algebra.
List of Publications
- S. Arumugam and M. Sundarakannan, On k-equivalence Domination in graphs, International Journal of Mathematical Combinatorics, 4(2010), 86-93.
- S. Arumugam and M. Sundarakannan, Equivalence Chain in graphs, J. Combin. Math. Combin. Comput., 80(2012),277-288.
- S. Arumugam and M. Sundarakannan, Equivalence Domination Number of a Tree, Proc. Jangjeon Math Soc., 15(2012) No. 3, 317-323.
- S. Arumugam and M. Sundarakannan, Equivalence Dominating Sets in Graphs, Utilitas Mathematica,. 91 (2013), 231-242.
- S. Arumugam and M. Sundarakannan, On equality in an upper bound for equivalence domination number, Quaestiones Mathematicae, 38(1), (2015):63-71.
- P. Deepa, P. Srinivasan and M. Sundarakannan, Local Coloring of Self Complementary Graphs, AKCE Int. Journal of Graphs and Combinatorics, 14(1), (2017) 35-41.
- Sundarakannan M., Arumugam S. Signed Cycle Domination in Planar Graphs. Lecture Notes in Computer Science(LNCS), vol 10398. Springer, Cham, (2017) pp 409-414.
Conference/ workshop Organized
- Workshop on Research Methodology and Latex at SSN College of Engineering, Chennai during April 19-20, 2013. (Convener)
- NBHM Sponsored National Seminar on Recent Developments in Algebraic Graph Theory at SSN College of Engineering, Chennai during July 05-06, 2013. (Coordinator)(Sponsored by NBHM)
- International Conference on Theoretical Computer Science and Discrete Mathematics (ICTCSDM- 2014) at SSN College of Engineering, Chennai during December 08-10, 2014. (Convener) (Sponsored by SSN Trust, DST, NBHM, CTS)
- Organized SSN CUBE OPEN 2017 in Association with World Cube Association (WCA) during 28.07.2017 to 29.07.2017.
Special/Invited Lectures Delivered
- Delivered a keynote address for the topic “Algebraic Concepts in Graph theory” in State level intercollegiate students seminar-Recent Advancement in Discrete Mathematics organized by Thiagarajar College, Madurai, Sep-23, 2011.
- Delivered a lecture for the topic “Typesetting in Latex” in the Instructional workshop on Latex organized by Maris Stella College, Vijayawada, Nov-19&20, 2011.
- Delivered a lecture for the topic “Spectrum of Graphs” in the Workshop on Cryptography and Graph Theory organized by St. Joseph College of Engineering, Sriperumbudur, Chennai, May 04, 2012.
- Delivered an invited talk on “Equivalence domination in graphs” in Instructional workshop on domination in graphs held at Kalasalingam University, Krishnan Koil during 30th January – 1st Feb , 2014.
- Delivered an invited talk on “A note on Local Coloring of graphs” in National Seminar on Graph Theory and its Applications held at St. Xavier’s College for Women, Aluva, Kerala during August 07 -09, 2014.
- Delivered an invited talk on “On cycle domination of weighted planar graphs” in International Conference on Graph Theory and its Applications at Amrita University, Coimbatore during 16 – 19, Dec 2015.
Books / Book Chapter Published
Book chapter published in the book titled
“Handbook of Graph Theory, Combinatorial Optimization, and Algorithms” by
Krishnaiyan “KT” Thulasiraman,
Subramanian Arumugam, Andreas Brandstädt, Takao Nishizeki
December 14, 2015 by Chapman and Hall/CRC Press
ISBN 9781584885955 – CAT# C5955
- Recognized as a research supervisor in Anna University, Chennai.
- Elected as an executive member for the period 2014 – 2016 from ACADEMY OF DISCRETE MATHEMATICS AND APPLICATIONS (ADMA).


A note from our editor in chief, Stephen Kiesling.
Many readers will remember the 1987 launch of James Gleick’s great book Chaos: Making of a New Science, and the idea that the flap of a butterfly’s wing can be the catalyst for a hurricane. At a lecture Gleick gave at the launch, what struck me was a video he showed on the mathematics of chaos theory. The video started with simple lines on the screen—the graph of an equation—that got more and more complex until what once looked like math now looked and moved like a living cell, as if chaos were an organizing principle of life.
Since then, we’ve learned how much chaos is actually inside us: those trillions of bugs that go about their own business and keep us alive. A human turns out to be a bus full of odd critters, and now we’re learning that cancer, too, is always on board. We “tend toward cancer, not away from it,” explains oncologist Dawn Lemanne MD. So the odd flap of a butterfly in the backseat of the bus can let loose a hurricane inside us.
What Maria Shriver calls “The Power of the Pause” (24) is to take a stand against chaos, moment to moment. Her extraordinary life has allowed he …


TOOTAY and ITRON have collaborated in an effort to conduct a technology demonstration of TOOTAY's Advanced 3D milling solutions to the international drilling community, in particular, the A & F International 2010 to be held in our hometown of Vancouver, Canada! It is part of an effort to showcase these new 3D milling solutions and share our new technologies with the world.
We comment on some topics concerning different classes of welded structures for the assessment of advanced statistical methods. The generalized inverse distribution family (GIDF) is a rich class of probability distributions that has been studied widely in a wide range of practical applications. This paper discusses the use of different sampling techniques for estimating the statistics of GIDF, namely the maximum likelihood, the exact methods and the least square methods. Weighted residual sampling methods are used for implementing the Monte Carlo and MCAL approaches as well as the Markov chain Monte Carlo (MCMC) methods. Then, an adaptive MCMC algorithm is proposed to improve the efficiency of the MCMC method and to reduce the computational burden. Furthermore, standard techniques for ranking alternatives within output space (SH and DSA) are also discussed with emphasis on the special cases in which only ordinal data are available. 7211a4ac4a


One of the things that I was dreading when I had kids was the day that they would come to me and need help with homework. I hated math at school and I mean hated it! By a country mile it was my least favorite subject and I really struggled with it. So I did a little homework myself to make sure that I could at least offer a helping hand to my kids. Thankfully they get math a heck of a lot better than I did. But I still want to share a few of the things I have used to help them with you guys.
They Need To Know The Concepts
There is no point in asking your child what is 5 + 2 if they do not have a proper comprehension of what add means. The key in the early days when getting your kid to understand math is making sure that they know what addition and subtraction is. Adding to and taking away are the terms that helped my son grasp it. We would use these little Batman toys and make addition and subtraction games with those and that was what eventually made him grasp it. For example my son would be Batman and he had these bad guy toys. I would get five bad guys and say if you put three in jail how many are left? And things like that.
Help Them With The Basics
By this I mean make it so they can answer simple math questions in just a few questions. Such simple questions like what is 2 + 3 and 4 – 2 and so on. Nothing really to brain taxing. The idea of this is to just get them ready and understand the basics of addition and subtraction and if you feel like they have that down add in a little multiplication and division questions as well. You want the questions to be easy enough so that they can get them quickly.
Help Them Write Numbers Neatly
Many schools say that around a quarter of the time when a child gets a math question wrong it’s because of sloppy hand writing. Take the time to make sure that they can write all the numbers properly and that they do not confuse numbers like a 6 and 9 which is what some children do. If their writing is nice and neat then they will have an easier time figuring out the questions.
Help Them Do Math Homework The Right Way
My youngest son was a real Johnny Cut Corners when it came to his homework. He would do all he could to get out of it or do it as quickly as possible. I had to show him the right way to do math homework as he was always getting it wrong, even though I knew he should be getting them right. The reason he was getting them wrong was that he was not reading the questions properly or writing down the working out of his answer. Once I got him to do that (which was no picnic let me tell you!) he would be able to do it much better.
You might also find helpful this related article: Ways To Motivate Your Child To Do Better In School


This will be the most fun you’ve had in a while – COME TO THIS CLASS!!!
Skill: 3 rounds of 2 attempts to throw a Slam-Ball as far as possible behind-the-head! (Think reverse caber toss!) Accomplished OUTSIDE unless its raining, and then please be careful of the lights! 🙂
20 minute Partner AMRAP:
P1 – Handstand Hold (Any variant, so long as your hands are closer to the ground than you head, and your head is closer to the ground than everything else!)
P2 – KB Swings, where the total math of (number of swings x KB Weight) equals 3x BW (Round Up!)
Every time you switch, you must *together* complete a number of burpees equal to the number of swings it took.
*Example: Kyle weighs 195#s. 195×3=585. If Kyle Swings a 2 pood KB (70#) then it will take him 585/70, or 9 swings (8.35 rounded to 9). Then the partners must complete a total of 9 burpees, and then switch. If Kyle had swung a 1.5 pood KB, he would have had to swing 585/53, or 12 swings (11.03 rounded to 12). and then partners would have to compete 12 burpees. If Kyle had only swung a .5 pood KB (18#), he would have had to do 33 swings and burpees!! Embrace the heavy KB!
*Score is Rounds complete + % of weight competed to the next round.


Price-earnings ratio, also called P/E ratio or referred to simply as PE, is a valuation method used to compare a company’s current share price to its per-share earnings.
PE is a simple number that’s easy to get from two numbers that can be found with the most basic of research. If you can do some basic math, you can calculate price-earnings ratio in your head and amaze your friends.
How to calculate price-earnings ratio
Before we get into the benefits of using price-earnings ratio to pick stocks, here’s an example of how it works:
First, find the current trading price for one share of stock in a company. Then find the earnings per share, or EPS. Divide the price per share by earnings per share and you have the price-earnings ratio:
Market value per share / Earnings per share (EPS) = Price-earnings ratio (PE)
EPS can be calculated a few different ways. It’s most often taken from the last four quarters — called a trailing PE. That can make a lot of sense because it’s for the most recent earnings period and show how much money a company is currently making.
Another EPS calculation is from the earnings expected in the next four quarters — called projected or forward PE. It doesn’t use actual data but is predicting possible outcomes for the stock.
A third method uses the sum of the last two actual quarters and the estimates of the next two quarters.
For sake of argument, we’ll use the trailing PE with earnings from the last four quarters. Here’s an example with real numbers:
If ABC company trades at $100 and has an earnings per share of $5, the calculation is: 100 / 5 = 20.
100 (share price) divided by 5 (EPS) equals a price-earnings ratio of 20.
What price-earnings ratio means
That’s a calculation you can do in your head, though it’s unnecessary with PE available online by looking up a stock price. But it’s what you can do with that number that matters.
While PE doesn’t paint the entire picture of a stock, it helps answer the basic question of if investing in the stock is worth the price. Just remember, trailing PE only tells you what happened in the past, and forward PE is an estimate of projected earnings.
PE is often used to decide if a stock is underpriced, overpriced or somewhere in the middle by comparing the price you pay for it to how much money the stock produces. Strong earnings are proof that a company is doing well, which will drive PE down.
What number is best?
A good rule of thumb when the evaluating price-earnings ratio is to look for a PE of 20, which is what the S&P 500 PE ratio is as of this writing in July 2015. What you’re normally looking for is a low PE.
If a PE for a stock goes up, it means the stock price has risen faster than earnings.
When researching a stock’s PE, take into account the PE of other stocks in that category. Yahoo, for example, breaks down businesses by sector and shows that technology stocks have an overall PE of 20.58. If the stock you’re looking at has a PE that’s double that, the stock may be overvalued.
PE can help determine if a company’s stock is selling for a fraction of what it’s really worth. That’s where the benefit comes in when determining if a stock is undervalued and worth buying.
A company can look overpriced if its PE is higher than the historical average for its sector. It can be underpriced if its PE is a lot lower than the average for its sector, such as among tech stocks.
If two stocks are the same price, for example, but one has a price-earnings ratio that’s half of the other, then the lower PE stock is a better deal because it offers twice the amount of earnings per share. The investor is getting a lot more for their money through higher earnings, even though the stock price is the same.


17 October 2012
7:00 PM to 8:00 PM
140 Grey Street , South Brisbane, QLD
Org: Queensland Conservatorium, Griffith University
Tel: (07) 3735 6241
Skip to Map
Concession $10.00, Adult $15.00. Tickets available at the door.
Join the Comprovisors again in October as they present their unique blend of technology, improvisation, mathematics and world music. Is this a shotgun wedding, or simply the reality in a musical world with no fixed boundaries?


Recognizing that God created every child with unique learning capabilities, Bethany Christian School utilizes skill-based grouping as an instructional model for the vital subjects of reading and math. This model entails assessing students and placing them in groups based on their achievement in these two central subjects while allowing students to be in an age-appropriate classroom with their grade level peers for all other subjects.
Reading is considered a paramount skill necessary for a happy, productive, and successful life. A child that is an excellent reader is a confident child, has a high level of self-esteem and is able to easily make the transition from learning to read to reading to learn.
Reading and analyzing texts helps your child to develop critical-thinking skills, such as observing, interpreting, and evaluating. When these components are present, your child will begin to make inferences about what is being read and be able to summarize key ideas and details, thus requiring your child to think critically. As this process develops, your child is able to apply these skills to real world situations. A solid reading foundation will help your child read independently and comprehend almost any type of text.
Math teaches logic and order. You can expect a mathematical equation to have a predictable outcome, and precise steps must be followed in order to attain that result. The discipline of mind that children develop in math class carries over into everyday life. A widely accepted premise is that math provides a vehicle through which critical-thinking skills are put into practice and refined. One example of mathematical critical thinking is when students are required to explain how they arrived at a solution to a complex problem or to describe the ideas behind a formula or procedure. Math intersects with reading. Concepts in math, such as averages, hypothesis, mean, and computation lend themselves to an increase in understanding, enhances vocabulary and complements effective comprehension.
Meeting students where they are in reading and math allows students to be instructed and challenged at an appropriate level of their capabilities without overwhelming them. Learning becomes fun, and the thrill of discovery spills into every other subject. This model has proven to be extremely effective for learners at all levels for well over a decade. One result of our distinctive model is that Bethany Christian School students consistently exceed state and national norms on standardized tests. Furthermore, students have consistently demonstrated they are at high school level math while still in middle school.
To learn more or to schedule a tour, please call our office at (480) 752-8993 or email us at firstname.lastname@example.org.


Hi, my name is Mark Haigh and I’m a lottery fan.
There I’ve said it. I’ve admitted my secret fascination with those little round numbered balls that tumble out of the big perspex bowl every week – and make someone a millionaire!
I’m also a bit of a science, maths and statistics fan.
Now many mathematics like to mock those who enjoy playing the lottery – you may have even heard the phrase ‘the lottery is a tax on the stupid’!
But not only is that deeply unfair, it’s also completely wrong.
Lottery players know they have a very small chance of winning. But quite simply the value of possibly becoming a millionaire overnight is worth way more to us than the value of our £1 coin (or $1 note, or whatever).
So yes cynical mathematicians of the world, you’re welcome to go out and smugly spend your £1 on something else, but we prefer the weekly excitement of those magic balls.
And the casual dream of what we would do with all that cash if it did actually happen to us.
And what else can you buy for a quid these days..?
Good luck, and happy dreaming,
P.S. If you want to get in touch about anything, head for my contact me page.
P.P.P.S. You can also follow me on Twitter here – send me a Tweet to say ‘hi’.


Dr. Rajkumari Mittal is faculty in the area of operations management at Lal Bahadur Shastri Institute of Management .She has invested 25 years of her career for teaching as well as research with students of both undergraduate and graduate level in many business and technology Institutes. She obtained her Ph.D. from the Mathematics and statistics department of Banasthali University. Her teaching interest exists for the subjects Statistics for Business, Operations Research, Production and Operations Management, Service Operations Management ,Project Management, Inventory management , and Supply chain Management. Her consultancy interests lie in Green Supply chain, Inventory management. Dr. Mittal has been credited with various publications in many coveted international and national journals.


The first fully-correct set of answers was submitted by Chuck Takahashi. Congrats, Chuck.
1. 636, since every competitor (except the ultimate winner) must lose once. The two finalists may or may not have played the same number of games, but this was not stated as one of the conditions.
2. 6 x pi meters. The length of the inner comm line is 2 x pi x r meters, while the length of the outer line is 2 x pi x (r + 3) meters. The difference is 6 x pi meters. All the other numbers (distance from equator, diameter of sphere, etc.) are extraneous factoids.
3. 1 fignet is 4000 km. The formula for a sphere's volume is 4/3 x pi x r (cubed), where r is radius; that for a sphere's surface area is 4 pi x r (squared). If we express the planet's radius in fignets and recall that its surface area in square fignets is equal to its volume in cubic fignets, we can determine the radius by equating the two formulae and solving for r. Thus, r = 3 (fignets). We know the diameter (24,000 km), so 1 fignet must be 4000 km.
4. Picard only gained 5 Lwaxana-less minutes. Since the shuttlecraft arrived 10 minutes early, it must have shaved off 5 minutes from each leg (starbase to asteroid, asteroid to starbase). The Enterprise had been moving from the asteroid 55 minutes (one hour minus 5 minutes). Since Picard was going to get rid of Lwaxana an hour after the Enterprise arrived at the asteroid, he only parted with her five minutes earlier (but it was probably worth it). The speed is irrelevant.
5. There are three Vulcan and one human survivors. If there are n survivors, of which x are Vulcans, the probability that two random survivors are Vulcans is [x (x-1)] / [n (n - 1)]. We are told this ratio is 1/2. Next, we must find integral values of x and n that give this expression a value of 1/2. The only values that allow a total of no greater than 14 are n = 4 and x = 3.
6. 120.7 km. Solution one: For simplicity, equate 1 unit with the space between ships (50 km), and the time it takes them to travel this distance. Their speed (distance/time) is also 1. Let x be the total distance traveled by the query and reply bozons, and also their speed. The speed of the query bozons relative to the ships is x - 1. The speed of the reply bozons is x + 1. The reply and query bozons both travel 1 unit and make the round trip in unit time, so: 1/ (x - 1) + 1/ (x + 1) = 1, which can be expressed as x (squared) - 2x - 1 = 0, for which the positive value of x is 1 + square root of 2. Multiply this by 50 km to get the final answer of 120.7 km. The warp speed is irrelevant.
Solution two: Rob Carvalho said this about solution one: "Your method is way wierd. Now I'm no expert in hyperrelativity, but if we were to treat this problem in a nonrelativistic way, it just makes sense to say that in the time that the Enterprise traveled 7/10 of its distance, the bozons traveled 17/10 that distance, and then while the Enterprise traveled the remaining 3/10, the bozons traveled back the 7/10 to meet the Addison. This means that the bozons traveled at about 2.42 times the speed of the ships (which is irrelevant, they will be assimilated), and thus went about 2.42 times the distance that the ships went, which is 120km. Why bring in higher math when simple conjecture/trial and error will do? But your answer is more precise."
Kevin O'Connell noted: "Bozons: how far in what frame of reference? The starships? Do we assume that space isn't warped around the ships as they travel at Warp 9? The answer will be different if measured from a "motionless" frame of reference (I suppose, since I don't know how warp speed affects the Lorentz transformation!). If observed from a planet being passed, the answer will further depend on the velocity of the planet.
Kevin, is, of course, totally correct. The given answer is from the starships' (or bozons') frame of reference.
7. Yes. Assume there are a maximum of n forehead wrinkles on one (very wrinkly) Klingon. If there are n Klingons, it is possible (though unlikely) that all have different numbers of wrinkles. However, since there are at least n + 1 Klingons, the n + 1st Klingon must share the same number of forehead wrinkles as one of the other Klingons.
8. Jake is 13. Let B = Benjamin's age, J = Jake's age, and F = Ben's father's age. His father is five times Jake's age, so F = 5J. When Benjamin is as old as his father is now, Jake will be F - B years older than he is now; his age will then be J + F - B. Since this is 8 years more than Benjamin's present age, J + F - B = B + 8. The third equation is F + B = 100. Solving these equations gives J = 13, B = 35 and F = 65.
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