A rational function can be written as a quotient of two polynomial expressions where the denominator has degree 1 or higher. In other words, there needs to be a variable in the denominator to be a rational function.
In this section, rational function models are used to solve work problems. A graphing calculator is needed.
Have you noticed that it takes a lot less time to get a job done when someone helps you than when you have to do the job alone? Rational functions are used to describe work functions.
Remember, the rate at which someone works multiplied by the amount of time it takes to do the work equals the amount of the job that is complete. When people work together the amount of the job each individual completes is added together to equal one job.
Mr. Jensen can wash and dry his car in 60 minutes. It takes his son, Peter, 45 minutes to wash and dry the car. Find an equation showing the amount of time it would take them to wash and dry the car if they worked together.
Let m represent the number of minutes they work together on the car.
The equation reflects Mr. Jensen’s part of the car plus Peter’s part of the car equaling the entire car being washed and dried.
Mr. Jensen + Peter = 1
one over 60 1 60 t + one over 45 1 45 t = 1
Another version of the equation:
t over 60 t 60 + t over 45 t 45 = 1
Put the above equation in your graphing calculator, this function represents a linear function rather than a rational function. What’s the difference between a linear and rational function—they both have fractions?
A rational function can be written as a quotient of two polynomial expressions where the denominator has degree 1 or higher. In other words, there needs to be a variable in the denominator to be a rational function.
Juan can mow and edge the lawn in 1.5 hours. When his sister, Manuela, helps him, they can mow and edge the lawn together in 45 minutes. How long would it take Manuela, working alone, to mow and edge the lawn?
Juan's work + Manuela's work = 1
1 over 90
1
90
(45) +
1 over x
1
x
(45) = 1
45 over 90
45
90
+
45 over x
45
x
= 1
The neighborhood pool uses two different-sizes of hoses to fill the pool. One hose can fill the pool in 8 hours. When both hoses work together, it takes 4.8 hours to fill the pool. Write an equation showing the time it would take the second hose to fill the pool alone.
Let x equal the hours it takes for the second hose to fill the pool alone. The rate of work for the second hose is 1/x.
The first hose fills 1/8th of the pool in one hour.
The equation is
one eighth 1 8 (4.8) + 1 over x 1 x (4.8) = 1 or 4.8 over 8 4.8 8 + 4.8 over x 4.8 x = 1
Jermaine can build a fence in 12 hours. If Jordan helps him, it takes them 7 hours to build the fence. Write a function rule to determine the time it takes Jordan to build the fence by himself.