8 Water is poured into an empty cone at a constant rate of \(8 \mathrm {~cm} ^ { 3 } / \mathrm { s }\)
After \(t\) seconds the depth of the water in the inverted cone is \(h \mathrm {~cm}\), as shown in the diagram below.
\includegraphics[max width=\textwidth, alt={}, center]{6ad3bac9-bf08-443d-8be2-b0c26209ffe8-10_468_481_497_778}
When the depth of the water in the inverted cone is \(h \mathrm {~cm}\), the volume, \(V \mathrm {~cm} ^ { 3 }\), is given by
$$V = \frac { \pi h ^ { 3 } } { 12 }$$
8
- Show that when \(t = 3\)
$$\frac { \mathrm { d } V } { \mathrm {~d} h } = 6 \sqrt [ 3 ] { 6 \pi }$$
8
- Hence, find the rate at which the depth is increasing when \(t = 3\)
Give your answer to three significant figures.
\(9 \quad\) Assume that \(a\) and \(b\) are integers such that
$$a ^ { 2 } - 4 b - 2 = 0$$