Answer
\[\begin{align}
& \mathbf{a}.\frac{dr}{dh}=-\frac{rh}{2{{r}^{2}}+{{h}^{2}}} \\
& \mathbf{b}.-\frac{6}{17} \\
\end{align}\]
Work Step by Step
\[\begin{align}
& A=\pi r\sqrt{{{r}^{2}}+{{h}^{2}}} \\
& \mathbf{a}\mathbf{.}\text{ For }A=1500\pi \\
& \pi r\sqrt{{{r}^{2}}+{{h}^{2}}}=1500\pi \\
& \text{Differentiate both sides with respect to }h \\
& \frac{d}{dh}\left[ \pi r\sqrt{{{r}^{2}}+{{h}^{2}}} \right]=\frac{d}{dh}\left[ 1500\pi \right] \\
& \pi r\frac{d}{dh}\left[ \sqrt{{{r}^{2}}+{{h}^{2}}} \right]+\sqrt{{{r}^{2}}+{{h}^{2}}}\frac{d}{dh}\left[ \pi r \right]=\frac{d}{dh}\left[ 1500\pi \right] \\
& \pi r\left( \frac{1}{2\sqrt{{{r}^{2}}+{{h}^{2}}}} \right)\left( 2r\frac{dr}{dh}+2h \right)+\pi \sqrt{{{r}^{2}}+{{h}^{2}}}\frac{dr}{dh}=0 \\
& \text{Solve for }\frac{dr}{dh} \\
& \frac{\pi {{r}^{2}}}{\sqrt{{{r}^{2}}+{{h}^{2}}}}\frac{dr}{dh}+\frac{\pi rh}{\sqrt{{{r}^{2}}+{{h}^{2}}}}+\pi \sqrt{{{r}^{2}}+{{h}^{2}}}\frac{dr}{dh}=0 \\
& \frac{\pi {{r}^{2}}}{\sqrt{{{r}^{2}}+{{h}^{2}}}}\frac{dr}{dh}+\pi \sqrt{{{r}^{2}}+{{h}^{2}}}\frac{dr}{dh}=-\frac{\pi rh}{\sqrt{{{r}^{2}}+{{h}^{2}}}} \\
& \left( \frac{\pi {{r}^{2}}}{\sqrt{{{r}^{2}}+{{h}^{2}}}}+\pi \sqrt{{{r}^{2}}+{{h}^{2}}} \right)\frac{dr}{dh}=-\frac{\pi rh}{\sqrt{{{r}^{2}}+{{h}^{2}}}} \\
& \left( \frac{\pi {{r}^{2}}+\pi {{\left( \sqrt{{{r}^{2}}+{{h}^{2}}} \right)}^{2}}}{\sqrt{{{r}^{2}}+{{h}^{2}}}} \right)\frac{dr}{dh}=-\frac{\pi rh}{\sqrt{{{r}^{2}}+{{h}^{2}}}} \\
& \left( \frac{\pi {{r}^{2}}+\pi \left( {{r}^{2}}+{{h}^{2}} \right)}{\sqrt{{{r}^{2}}+{{h}^{2}}}} \right)\frac{dr}{dh}=-\frac{\pi rh}{\sqrt{{{r}^{2}}+{{h}^{2}}}} \\
& \left( 2\pi {{r}^{2}}+\pi {{h}^{2}} \right)\frac{dr}{dh}=-\pi rh \\
& \frac{dr}{dh}=-\frac{\pi rh}{2\pi {{r}^{2}}+\pi {{h}^{2}}} \\
& \frac{dr}{dh}=-\frac{rh}{2{{r}^{2}}+{{h}^{2}}} \\
& \\
& \mathbf{b}.\text{ Evaluate when }r=30\text{ and }h=40 \\
& \frac{dr}{dh}=-\frac{\left( 30 \right)\left( 40 \right)}{2{{\left( 30 \right)}^{2}}+{{\left( 40 \right)}^{2}}} \\
& \frac{dr}{dh}=-\frac{1200}{2\left( 900 \right)+\left( 1600 \right)} \\
& \frac{dr}{dh}=-\frac{6}{17} \\
\end{align}\]
