Q. 54 - Precalculus Enhanced with Graphing Utilities

Question

An economy car rented in Florida from
National Car Rental on a weekly basis costs $\backslash (95$ per week. Extra days cost $\backslash )24$ per day until the day rate exceeds the weekly rate, in which case the weekly rate applies. Also, any part of a day used counts as a full day. Find the cost $C$ of renting an economy car as a function of the number $x$ of days used, where $7...x...14$. Graph this function.

Step-by-Step Solution

Answer

The cost function for the renting an economy car is given by,

$C\left(x\right)=\left\{\begin{array}{l}24x-73, 7\le x\le 11\\ 13.57x+41.71, 11<x\le 14\end{array}\right.$

The graph of the function is represented as:

Step 1. Given

The charge for weekly basis is $\$95$ per week.

Extra days cost $\$24$ per day until the day rate exceeds the weekly rate, in which case the weekly rate applies.

To find the cost $C$ of renting an economy car as a function of the number $x$ of days used, where $7...x...14$

To graph the function.

Step 2. Find the cost function

From the given,

for $x>7$, the extra days exceeds the weekly days after four days, so for $7\le x\le 11$, the cost function is,

$$\begin{gathered} C\left(x\right)=95+24(x-7) \\ =95+24x-168 \\ =24x-73 \end{gathered}$$

Step 3. Find the cost function

The cost function for $11<x\le 14$ is,

$$\begin{gathered} C\left(x\right)=95+4\left(24\right)+\frac{95}{7}(x-11) \\ =95+96+\frac{95x}{7}-\frac{1045}{7} \\ =191+\frac{95x}{7}-\frac{1045}{7} \\ =13.57x+41.71 \end{gathered}$$

Step 4. The cost function

The cost function can be expressed as,

$C\left(x\right)=\left\{\begin{array}{l}24x-73, 7\le x\le 11\\ 13.57x+41.71, 11<x\le 14\end{array}\right.$

Step 5. Graph the function

The graph of the function is represented as: