1.24: Figure 4 shows a container in the shape of a right circular cone.

4PM1/1R/June/2023 — Question 6 · 11 marks

1.24 diagram 1
Figure 4 shows a container in the shape of a right circular cone.
The container is fixed with its axis of symmetry vertical.
The vertical angle of the container is 60∘\displaystyle 60^\circ as shown in the diagram.
At time t\displaystyle t seconds, t>0\displaystyle t > 0, the height of oil in the container is h\displaystyle h cm and the volume of oil in the container is Vcm3\displaystyle V \mathrm{cm}^3
(a) Show that V=19πh3\displaystyle V = \frac{1}{9} \pi h^3
(3)
At time t\displaystyle t seconds the surface area of oil in the container is Acm2\displaystyle A \mathrm{cm}^2, as shown in Figure 4
Oil is dripping out of the bottom of the container at a constant rate of 4cm3/s\displaystyle 4 \mathrm{cm}^3/\mathrm{s}.
(b) Find the exact rate of change, in cm2/s\displaystyle \mathrm{cm}^2/\mathrm{s}, of the surface area of oil in the container when h=24\displaystyle h = 24
(8)