The monthly cost of driving a car depends on the number of miles driven. Julia found that in October it cost her $\$ 200$ to drive $300 \mathrm { mi }$ and in July it cost her $\$ 350$ to drive $600 \mathrm { mi }$. Express the monthly $\operatorname { cost } C$ as a function of the distance driven $d$ assuming that a linear relationship gives a suitable model.
Select the correct answer.
A) $C = - 50 d + 0.5$
B) $C = 50 d - 0.5$
C) $C = 0.5 d + 50$
D) $C = 2 d + 50$
E) $C = 0.5 d - 50$

Question

Quizplus · Accepted Answer

First, we notice that we can use the formula for the equation of a line, $y=mx+b$, where $y$ represents the cost, $x$ represents the distance driven, $m$ represents the slope, and $b$ represents the $y$-intercept.

Next, we can determine the slope of the line using the two points given:

\begin{align*}
m &= \frac{\text{change in } y}{\text{change in } x} \
&= \frac{\$350 - \$200}{600 \mathrm{mi} - 300 \mathrm{mi}} \
&= \frac{\$150}{300 \mathrm{mi}} \
&= \frac{1}{2} \frac{\$}{\mathrm{mi}}
\end{align*}

This tells us that for every mile driven, the cost increases by 50 cents.

Finally, we can use this slope and one of the points to find the $y$-intercept:

\begin{align*}
\$200 &= \frac{1}{2} \frac{\$}{\mathrm{mi}} (300 \mathrm{mi}) + b \
b &= \$50
\end{align*}

Therefore, our equation is:

$$C = 0.5d + 50 $$

The Monthly Cost of Driving a Car Depends on the Number