All solutions here are SUGGESTED. Mr. Teng will hold no liability for any errors. Comments are entirely personal opinions.

(i)
Volume, $V = \pi r^2 h + \frac{2}{3} \pi r^2 = k$

$\Rightarrow h = \frac{3k - 2\pi r^3}{3 \pi r^2}$

Surface Area, $A = 2 \pi r^2 + 2 \pi rh + \pi r^2$

$= 3 \pi r^2 + 2 \pi r \frac{3k - 2\pi r^3}{3 \pi r^2}$

$= \frac{5 \pi r^2}{3} + \frac{2k}{r}$

$\frac{dA}{dr} = \frac{10 \pi r}{3} - \frac{2k}{r^2}$

Set $\frac{dA}{dr} = 0$

$\Rightarrow r = (\frac{3k}{5 \pi})^{\frac{1}{3}}$

$\frac{d^2A}{dr^2} = \frac{10\pi}{3} + \frac{4k}{r^3} = 10 \pi > 0$ when $r = (\frac{3k}{5 \pi})^{\frac{1}{3}}$.

Thus A is minimum when $r = (\frac{3k}{5 \pi})^{\frac{1}{3}}$

$\Rightarrow h = r = (\frac{3k}{5 \pi})^{\frac{1}{3}}$

(ii)
Surface Area, $A = \frac{5 \pi r^2}{3} + \frac{400}{r} = 180$

$\Rightarrow 5\pi r^3 + 1200 - 540 r = 0$

Using the Graphing Calculator, we find that $r = 3.037205 \mathrm{~or~} 3.72153$

$h = 4.88 \mathrm{~or~} 2.12$

Since $r \textless h, \mathrm{~then~} r = 3.04, h = 4.88$