Question

A rental car company rents cars for a flat fee of $20$ an hourly charge of $10.25$. Therefore, the total cost $C$ to rent a car is a function of the hours $t$ the car is rented plus the flat fee.

a. Write the formula for the function that models this situation.
b. Find the total cost to rent a car for $2$ days and $7$ hours.
c. Determine how long the car was rented if the bill is $\$432.73$.

Answer 1

Step- 1: Write the formula for the function that models this situation:

Let $t$ be the number of hours a car is rented for.

Hence, the cost function $C\left(t\right)$ becomes: $C\left(t\right)=20+10.25t$.

Therefore, the formula for the function is $C\left(t\right)=20+10.25t$.

Step- 2:Find the total cost to rent a car for $\mathbf{2}$ days and $\mathbf{7}$ hours:

$2\mathrm{days}=24\times 2=48\mathrm{hours}$

Hence, the total number of hours, $\left(t\right)=48+7=55\mathrm{hours}$.

Rent for $55$ hours, substitute $t=55$ in the obtained function:

$\begin{array}{rcl}C\left(55\right)& =& 20+10.25\times 55\\ & =& 583.75\end{array}$.

Therefore, the total cost is$\mathbf{\$}\mathbf{583}\mathbf{.}\mathbf{75}$.

Step- 3: Determine how long the car was rented if the bill is$\mathbf{\$}\mathbf{432}\mathbf{.}\mathbf{73}$.

It is given that the bill is $\$432.73$.

So:

$$\begin{gathered} C\left(t\right)=432 \\ \Rightarrow 20+10.25t=432.73 \\ \Rightarrow 10.25t=432.73-20 \\ \Rightarrow 10.25t=412.73 \\ \Rightarrow t=\frac{412.73}{10.25} \\ \Rightarrow t=40.27 \end{gathered}$$

Therefore, the car was rented for $\mathbf{40}\mathbf{.}\mathbf{27}\mathbf{}\mathbf{hours}$.