A cylinder X has radius, $r$ cm and height $h$ cm. A hemisphere S is such that its radius is half of the radius of cylinder X.

1. Write the radius of the hemisphere in terms of $r$ and form its volume

Radius of hemisphere S $=\dfrac{r}{2}$.

Volume of hemisphere
$=\dfrac{2}{3}\pi\left(\dfrac{r}{2}\right)^3$

2. Use the condition that the cylinder volume is 4 times the hemisphere volume

Volume of cylinder X $=\pi r^2h$.

Given cylinder volume $=4\times$ hemisphere volume,
$\pi r^2h=4\left[\dfrac{2}{3}\pi\left(\dfrac{r}{2}\right)^3\right].$

3. Simplify and solve for $h$

$\pi r^2h=4\left[\dfrac{2}{3}\pi\cdot\dfrac{r^3}{8}\right]$
$\pi r^2h=\dfrac{1}{3}\pi r^3$
$h=\dfrac{\frac{1}{3}\pi r^3}{\pi r^2}=\dfrac{r}{3}.$

1. Find the linear scale factor from cylinder X to cylinder Y

For geometrically similar solids,
$\text{area ratio}=\text{linear scale factor}^2.$
Given curved surface area ratio
$X:Y=9:4,$
so the linear ratio is
$X:Y=\sqrt{9}:\sqrt{4}=3:2.$

2. Use the height of cylinder X from part (a) to find the height of cylinder Y

From part (a), the height of cylinder X is
$h=\frac{r}{3}.$
Since the linear ratio $X:Y=3:2$,
$\text{height of Y}=\frac{2}{3}\times \frac{r}{3}=\frac{2r}{9}.$