Did you know that a rational function can also represent inverse variation?

For example, a rational function is a polynomial divided by another polynomial.

y = 20/x is a polynomial function, but it's also inverse variaton.

y = k/x reprsents inverse variation, y varies inversely as x and k is the constant.

In the above example, the constant is 20.

Consider these two problems involving volume.

The diameter of a baseball is approximately 7.37 cm. A hockey puck is cylindrical with a thickness

of 2.54 cm and a diameter of 7.6 cm. Which has the greatest volume?

Solution:

A baseball is spherical and the volume of a sphere is (4/3)πr³. The radius of the sphere is half the diameter, so

r = 3.685 cm. Using 3.14 for π, the volume of the baseball is (4/3)(3.14)(3.685)³ = 209.5 cm³. (rounded to one decimal place)

The volume of a cylinder is πr²h. The radius of the hockey puck is 3.8 cm, the height is 2.54 cm. Therefore the volume of the hockey puck is (3.14)(3.8)²(2.54) = 115.2 cm³. (rounded to one decimal place)

The volume of a baseball is nearly double the volume of a hockey puck.

Suppose you have a metal cone shaped container and a plastic container in the shape of a shoe box. The cone shaped container is 14 inches high with a diameter of 10 inches. The plastic container is 12 inches long, 6 inches wide and 4 inches high. You want to fill up one container with water. Which container will hold the most water?

Solution:

The volume of a cone is (1/3) πr²h.

Therefore, the volume of the cone shaped container is (1/3)(3.14)(5)²(14) = 366.3 inches³ (rounded to one decimal place)

The volume of a rectangular solid is length times width times height, therefore the volume of the plastic container is (12)(6)(4) = 288 inches³.

The metal cone shaped container will hold the most water.

What is 2 X 2 + 2 X 2 + 2 - 2 X 2 ?

Be careful with a problem like this. People might say there are several answers depending on what operation is performed in what order. That's why there are the order of operations to follow.

By following the order of operations, the answer is 6. Follow the order of operations and perform from left to right. Multiplication comes before addition and addition comes before subtraction.

4 + 4 + 2 - 4
10 - 4
6

The time it takes a home building company to build a new home is directly proportional to the size of the home and inversely proportional to the number of people on the building crew. If it takes 10 workers 4 weeks to build a 2,000 square foot home, how long will it take 6 workers to build the home?

Solution:

Let t = time it takes to build a home

w = number of workers

s = size of home

The joint variation between these variables can be represented by

t = ks/w

Substitute 4 for t, 2000 for s and 10 for w and solve for k.

4 = k(2000)/10

Multiply both sides by 10 to clear the denominator.

40 = k(2000)

Divide by 2000 to get

40/2000 = 1/50 = k.

Now substitute 6 for w, 2000 for s and 1/50 for k and solve for t.

t = (1/50)(2000/6)

t = 2000/300, t = 6.66. Therefore, it takes 6 workers 6 2/3 weeks to build a 2,000 square foot home.

Consider the following problem:

The instructions on a can of paint read that a half gallon of paint will cover approximately 175 square feet. How many half gallon cans of paint must be purchased to apply 2 coats of paint to cover 600 square feet per coat?

Solution:

We can think of this problems in terms of direct variation. The amount of paint needed is directly proportional to the area that needs to be painted.

Let y = amount of paint in gallons

x = area to be painted in square foot.

Therefore, we use the equation for direction variation, y = kx.

Substitute ½ for y and 175 for x and solve for k.

½ = k(175) (Divide both sides by 175)

1/350 = k

To apply 2 coats over 600 square feet means we need enough paint to cover 1,200 square feet, therefore

y = (1/350)(1200)

y = 3.42 gallons.

We need to purchase 7 cans of paint because 7(1/2) = 3.5.

Consider the following problem:

The flight of two planes is being tracked on a rectangular coordinate system. The flight of the first plane is defined by the equation 3x + 4y = 12 and the flight of the second plane is defined by the equation y = -(3/4)x + 10. If the planes continue along the same paths, is there any danger of them colliding? (Assume the planes are flying at the same altitude)

Solution:

To determine if the planes will collide, we need to find the point of intersection of the two lines.

Substitute -(3/4)x + 10 for y in the first equation and solve for y.

3x + 4[-(3/4)x + 10] = 12

3x - 3x + 40 = 12

40 = 12 is a false statement. Therefore, there is no solution. There is no chance of the two planes colliding if they keep flying along the same paths.

Another way to solve this is to get the slope of each line. If the slopes are the same without the same y – intercept, then the lines are parallel and therefore the planes cannot collide.

3x + 4y = 12

4y = 12 - 3x

y = 4 - (3/4)x. Notice the slope of both lines is -3/4. The y – intercepts are different. Therefore there is no chance of the planes colliding if they continue along their same path.

Here's the preface and about the author section of my second book. If anyone is interested in the book, let me know.

Preface

Algebra Simplified Intermediate & Advanced picks up where my first book, Algebra Simplified Basic & Intermediate left off. It is intended to assist students in intermediate and advanced topics studied in a 2^nd year high school algebra course or an intermediate college algebra course. The material is presented in textbook style format with each concept illustrated through numerous examples. The examples are solved methodically to explain each concept as simply as possible. Important notes and tips for easier learning are presented in bold throughout the book. The goal is provide readers sufficient detail in the examples so they can solve similar problems on their own, which are presented at the end of each section.

Specific topics covered in this book include division of polynomials, finding roots of polynomials, factoring techniques such as quadratic formula and completing the square, radicals, rational exponents, complex numbers, logarithms, conic sections, composition of functions, inverse functions, arithmetic and geometric sequences, matrices and basic probability and counting techniques At the end of each chapter there is a section titled Key Terms and Concepts to Review. Answers to problem sets along with key steps in the solution of each problem are also found at the end of each chapter. The book concludes with a glossary of terms used throughout the book.

About the Author

Kerry Kauffman has a Bachelor of Science Degree in Statistics from Lehigh University. He has tutored students in algebra, geometry, trigonometry, statistics, calculus, number theory and other courses at the high school and college level since 2000. He has worked on tutoring websites tutorsteach.com, tutornation.com and liveperson.com and has assisted students taking online college courses from Harvard University, Strayer University, Kaplan University and University of Phoenix among others. He is the author of Algebra Simplified Basic & Intermediate, which was published in 2011.