Chapter 4.7, Problem 21PPS

a.

To determine

To find: The inverse of function $C(x)$ .

Answer to Problem 21PPS

  $C^{-1}(x)=\frac{x-25}{45}$

Explanation of Solution

Given information: Total amount collected is $C(x)=25+45x$

Calculation:

Let $C(x)=25+45x$ as

  $y=25+45x$

Swap the variables,

  $x=25+45y$

Solve the equation for ‘y’ that would be our inverse.

Subtract both sides 25.

  $x-25=45y$

Divide both sides by 45.

  $\frac{x-25}{45}=\frac{45y}{45}$

  $\frac{x-25}{45}=y$

The inverse function is (replace y as $C^{-1}(x)$ )

  $C^{-1}(x)=\frac{x-25}{45}$

b.

To determine

To find: The quantities represented by x and $C^{-1}(x)$ in this context.

Answer to Problem 21PPS

In the inverse function x represents the total amount collected and the $C^{-1}(x)$ represents the number of times Chunk mows the Foster’s lawn.

Explanation of Solution

Given information: Total amount collected is $C(x)=25+45x$ and the x represents number hours Chunk mow the Foster’s lawn.

While calculating inverse of a function, all x are replaced with $C(x)$ and all $C(x)$ replaced with x. Then solve it for $C(x)$ and replace $C(x)$ as $C^{-1}(x)$ .

c.

To determine

To find: The number of times Chunk mow the Foster’s lawn when he collects $1015.

Answer to Problem 21PPS

Chunk mows the Foster’s lawn for 22 times.

Explanation of Solution

Given information:

Calculation:

Let’s plug in $C(x)$ as $1015. Then solve the equation for x.

  $1015=25+45x$

Subtract both sides 25.

  $990=45x$

Divide both sides by 45.

  $x=22$

Conclusion: Chunk mows the Foster’s lawn for 22 times.