A solid cylinder has radius $r \mathrm{~cm}$, height $h \mathrm{~cm}$ and total surface area $900 \mathrm{~cm}^{2}$. Find the exact value of the maximum possible volume of the cylinder. Find also the ratio $r: h$ that gives this maximum volume.
[7]
1:2
3. Total surface area $=900$
$2 \pi r^2+2 \pi r h=900$
$2 \pi r l=900-2 \pi r^2$
$h=\frac{900-2 \pi r^2}{2 \pi r}$
Volume, $V=\pi r^2 h$
$=\pi r^2\left(\frac{900-2 \pi r^2}{2 \pi r}\right)$
$V=450 r-\pi r^3$
$\frac{\mathrm{d} V}{\mathrm{~d} r}=450-3 \pi r^2$
At stationary point,
$\frac{\mathrm{d} V}{\mathrm{~d} r}=0$
$450-3 \pi r^2=0$
$r^2=\frac{450}{3 \pi}$
$r=\sqrt{\frac{150}{\pi}}$ or $r=-\sqrt{\frac{150}{\pi}}$ (N.A., $\left.r \geqslant 0\right)$
When $r=\sqrt{\frac{150}{\pi}}$
$\begin{aligned} V &=450 r-\pi r^3 \\ &=450\left(\sqrt{\frac{150}{\pi}}\right)-\pi\left(\sqrt{\frac{150}{\pi}}\right)^3 \\ &=\sqrt{\frac{150}{\pi}}\left[450-\pi\left(\frac{150}{\pi}\right)\right] \\ &=\sqrt{\frac{150}{\pi}}(300) \\ V &=1500 \sqrt{\frac{6}{\pi}} \\ \frac{V}{\pi} & \frac{1500 \sqrt{6 \pi}}{\pi} \\ \frac{\mathrm{d}^2 V}{\mathrm{~d} r^2} &=-9 \pi r \end{aligned}$
When $r=\sqrt{\frac{150}{\pi}}$,
$\frac{\mathrm{d}^2 V}{\mathrm{~d} r^2}<0(V$ is maximum $)$
$h=\frac{900-2 \pi r^2}{2 \pi r}$
$=\frac{900-2 \pi\left(\sqrt{\frac{150}{\pi}}\right)^2}{2 \pi\left(\sqrt{\frac{150}{\pi}}\right)}$
$=\frac{900-2 \pi\left(\frac{150}{\pi}\right)}{2 \pi\left(\sqrt{\frac{150}{\pi}}\right)}$
$\begin{aligned} \frac{r}{h} &=\frac{\sqrt{\frac{150}{\pi}}}{\left[\frac{900-300}{2 \pi\left(\sqrt{\frac{150}{\pi}}\right)}\right]} \\ &=\frac{\sqrt{\frac{150}{\pi}}\left(\sqrt{\frac{150}{\pi}}\right)(2 \pi)}{600} \\ &=\frac{\frac{150}{\pi}(2 \pi)}{600} \\ &=\frac{1}{2} \\ \therefore r: h=1: 2 \end{aligned}$