Differentiation

An inverted cone has height \(12\text{ cm}\) and rim radius \(6\text{ cm}\).
Water enters at \(12\pi\text{ cm}^3/\text{s}\). When the water depth is \(h\) cm, water leaks out at \(\pi h\text{ cm}^3/\text{s}\).

1. Show that \( \frac{\mathrm{d}h}{\mathrm{d}t}=\frac{4(12-h)}{h^2}, 0<h<12. \)

2. Find \(\dfrac{\mathrm{d}h}{\mathrm{d}t}\) when \(h=4\) and when \(h=8\).

3. Explain why doubling the depth does not simply halve its rate of increase.