$\text { If the annual rate of inflation averages } 8 \% \text { over the next } 10 \text { years, the approximate }$ cost $C$ of goods or services during any year in that decade is $C ( t ) = P ( 1.08 ) ^ { t }$ where $t$ is the time in years and $P$ is the present cost. The price of an oil change for your car is presently $\$ 23.65$. Find the rate of change of $C$ with respect to $t$ when $t = 3$. Round your answer to three decimal places.
A) $32.176$ dollars per year
B) $1.966$ dollars per year
C) $2.293$ dollars per year
D) $5.962$ dollars per year
E) $32.730$ dollars per year

Question

Quizplus · Accepted Answer

We need to find the derivative of the given function with respect to $t$ and then evaluate it at $t=3$. 
\begin{align*}
\frac{d}{dt}C(t) &= \frac{d}{dt} \left(P(1.08)^tight) \
&= P(1.08)^t \ln(1.08) \
\end{align*}
Evaluating at $t=3$ and substituting $P=23.65$, we get
\begin{align*}
\frac{d}{dt}C(3) &= 23.65 (1.08)^3 \ln(1.08) \
&\approx \boxed{2.293} 	ext{ dollars per year}
\end{align*}

If the annual rate of inflation averages 8% over the next 10 years, the approximate \text { If the annual rate of inflation averages } 8 \% \text { over the next } 10 \text { years, the approximate } If the annual rate of inflation averages 8% over the next 10 years, the approximate

$\text { If the annual rate of inflation averages } 8 \% \text { over the next } 10 \text { years, the approximate }$