Single Variable Calculus, Chapter 1, 1.2, Section 1.2, Problem 18

The charges of driving a car mostly depends on the distance it was driven. Suppose that Harley found that in May it cost him \$380 to drive 480 miles and in June it cost him \$460 to drive 800 miles.


a.) Assuming that a linear model fits on the given data, express the monthly cost C as a function of the distance driven d.


$
\begin{equation}
\begin{aligned}
c &= ad + k \text{ where:}\\
c & = \text{monthly cost in dollars}\\
a &= \text{slope}\\
d &= \text{distance driven in miles}\\
k &= \text{other expenses}
\end{aligned}
\end{equation}
$



Given:



$
\begin{equation}
\begin{aligned}
\text{when } c &= 380, \, d = 480\\
c &= 460, \, d = 800\\
380 &= 480a + k && \text{Equation 1}\\
460 &= 800a + k && \text{Equation 2}
\end{aligned}
\end{equation}
$


*Combining equations 1 and 2


$
\begin{equation}
\begin{aligned}
a & = \displaystyle \frac{1}{4}\\
k & = 260
\end{aligned}
\end{equation}\\
\boxed{.: c = \displaystyle \frac{d}{4}+260}
$


b.) Predict the cost of driving 1500 miles per month by using the model in part (a).


$
\begin{equation}
\begin{aligned}
\displaystyle c &=\frac{d}{4} + 260; \text{ when } d= 1500\\
\displaystyle c &=\frac{1500}{4} +260\\
c & = \$635
\end{aligned}
\end{equation}
$


c.) Draw the graph of the linear function then state what does the slope represent.



The slope is $\displaystyle \frac{1}{4}$, it represents the change of cost
for every change of distance.