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# A study of driving costs based on 2012 medium-sized sedans found that the average cost (car payments, gas, insurance, upkeep, and depreciation) in cents per mile is approximately $C(x)=\frac{1910.5}{x^{1.72}}+42.9 \ (5 \leq x \leq 20)$ where x (in thousands) denotes the number of miles the car is driven each year. Show that C is a decreasing function of x on the interval (5, 20). What does your result tell you about the average cost of driving a 2012 mediumsized sedan in terms of the number of miles driven?

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It is given that the average cost in cents per mile is approximately

$C(x) = \frac{1910.5}{x^{1.72}} + 42.9\ \ \ \ \ \ (5 \leq x \leq 20)$

where $x$ (in thousands) denotes the number of miles the car is driven in each year.

The function $C$ can be rewritten as

$C(x) = 1910.5x^{-1.72} + 42.9\ \ \ \ \ \ (5 \leq x \leq 20)$

The derivative of $C(x)$ then, is

\begin{align*} C'(x) & = 1910.5(-1.72)x^{-2.72}\\ & = -3286.06x^{-2.72}\\ & = -\frac{3286.06}{x^{2.72}} \end{align*}

Since $5 \leq x \leq 20$, we obtain $C'(x) < 0$ for all $x$ and therefore, we can say that $C$ is a decreasing function of $x$ in the interval $(5,20)$.

It means that the average cost of driving a 2012 medium-sized sedan decreases as the number of miles driven increases. This is shown in the figure below.

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