1.53: Figure 2 shows a hollow right circular cone fixed with its axis of symmetry vertical.

4PM1/1/November/2025 — Question 9 · 13 marks

1.53 diagram 1
Figure 2 shows a hollow right circular cone fixed with its axis of symmetry vertical.
The vertical angle of the cone is 60∘\displaystyle 60^\circ.
Initially the cone is empty.
At time t=0\displaystyle t=0 liquid starts to fill the cone at a constant rate of 0.2 cm3/s\displaystyle 0.2\,\mathrm{cm}^3/\mathrm{s}.
At time t\displaystyle t seconds after the liquid starts to fill the cone, the height of the liquid is h\displaystyle h cm above X\displaystyle X.
(a) Show that
h=9t5π3.h=\sqrt[3]{\frac{9t}{5\pi}}.
(5)
The surface area of the liquid, shown shaded in Figure 2, is increasing at a constant rate of p cm2/s\displaystyle p\,\mathrm{cm}^2/\mathrm{s} when t=6\displaystyle t=6.
(b) Find, to 3 significant figures, the value of p\displaystyle p.
(8)