1.45: Past-paper question 7

Figure 2 shows a hollow hemisphere with radius 20 cm

The hemisphere contains liquid, which is dripping out of a small hole at the lowest point $\displaystyle A$ at a constant rate of $\displaystyle k \mathrm{cm}^3/\mathrm{s}$

At time $\displaystyle t$ seconds after the liquid starts to drip from the hemisphere, the height of the liquid is $\displaystyle h$ cm above $\displaystyle A$

The volume $\displaystyle V \mathrm{cm}^3$ of liquid in the hemisphere is given by

When $\displaystyle h = 12$, the height of the liquid is decreasing at a rate of $\displaystyle \frac{1}{60} \mathrm{cm}/\mathrm{s}$

Find the value of $\displaystyle k$

Give your answer in terms of $\displaystyle \pi$