What is one real world example on multiplying and dividing rational expressions?

Question

I am doing a math booklet on multiplying and dividing rational expressions for incoming Algebra 2 students. The contents of the book must contain a cover page, 2 step by step problems solved, 10 practice problems with worked out solutions, and one real world example about the topic. I have done everything except the real world example, which is what  I am stuck on. I have no idea who would use this, besides math teachers. My algebra teacher gave me this example:

 

"Professional football players would most likely use quadratic functions to help them make plays."

 

This assignment is due on Tuesday (05/30/17) please help me. I have to print this booklet out before then. 

Answer 1

Here is one:

 

A cannonball is fired straight up at a speed of 80 ft/s from the bottom of a crater.  The bottom of the crater is 96 feet below ground level.  You observed that the cannonball reached ground level on the way up 2 seconds after firing.  Based on this observation, find how many seconds after firing it reached ground level again, on the way down.

 

Hint:

 

y(t) = -(1/2)gt² + v_0yt + y₀

g = 32 ft/s²

 

Solution:

 

v_0y = 80 ft/s

Let y = 0 be the ground level.  

y₀ = -96 ft

 

y(t) = -16t² + 80t - 96, t in seconds, y in feet

 

At ground level:

 

y = 0

-16t² + 80t - 96 = 0

 

Divide through by -16 to simplify and get

 

t² - 5t + 6 = 0

 

Since y = 0 at 2 seconds, we know that t = 2 is a root of

 

t² - 5t + 6 = 0.

 

Let t₁ = the time the cannonball reaches ground level on the way down.  Then y(t₁) = 0, and t₁ is also a root.  Then

 

t² - 5t + 6 = (t - 2)(t - t₁)

 

t - t₁ = (t² - 5t + 6) / (t - 2) = [(t - 2)(t - 3)] / (t - 2) = t - 3  [Here is use of a rational expression.]

 

t - t₁ = t - 3

 

t₁ = 3

 

It reaches ground level on the way down 3 seconds after being fired up.

 

 

 

Answer 2

First off, quadratic functions are not rational functions.  Rational expressions are in the form of fractions.  An expression divided by another expressions.  Quadratics are continuous functions, whereas rationals have restrictions. An example of a rational of expression is

 

h(x) = f(x) / g(x)

 

where:
h(x) is the rational expression

g(x) and g(x) are two separate functions

 

h(x) has restrictions at x that make g(x)=0

 

 

In the case of quadratic functions, professional football players would use them to determine the range that a football is thrown or kicked across the field.   They can also use them to find how high a player can throw the football.