Answer
$ a.\quad \$ 101.22$
$ b.\quad \$ 111.86$
$ c.\quad \$ 117.59$
Work Step by Step
The rate of change of $A$ with respect to $r$ is $\displaystyle \frac{dA}{dr}$.
To find $\displaystyle \frac{dA}{dr}$, we use the chain rule
\begin{align*}
A(r)&=1500(1+\displaystyle \frac{r}{36,500}]^{1825} \quad & \\\\
\displaystyle \frac{dA}{dr}&= 1500\cdot 1825 [1+\displaystyle \frac{r}{36,500}]^{1824}\cdot [1+\frac{r}{36,500}]^{\prime}\quad & \\
&= 1500\cdot 1825 [1+\displaystyle \frac{r}{36,500}]^{1824}[\frac{1}{36,500}] \quad & \\
&=\displaystyle \frac{ 1500\cdot 1825}{36,500} [1+\frac{r}{36,500}]^{1824}\quad & \\
&=75[1+\displaystyle \frac{r}{36_{9}500}]^{1824}
\end{align*}
(a)
For $r=6\%,$
$\displaystyle \frac{dA}{dr}=75(1+\frac{6}{36,500}]^{1824}= \$ 101.22.$
(b)
For $r=8\%,$
$\displaystyle \frac{dA}{dr}=75(1+\frac{8}{36,500}]^{1824}= \$ 111.86.$
(c)
For $r=9\%,$
$\displaystyle \frac{dA}{dr}=75(1+\frac{9}{36,500}]^{1824}= \$ 117.59.$
