Problem

A regular icosahedron of side length $x$ cm, shown in Figure 1, is expanding uniformly.

The icosahedron consists of 20 congruent equilateral triangular faces of side length $x$ cm.

(a) Show that the surface area, $A$ cm$^2$, of the icosahedron is given by

$$A=5\sqrt{3}x^2$$

Given that the volume, $V$ cm$^3$, of the icosahedron is given by

$$V=\frac{5}{12}\left(3+\sqrt{5}\right)x^3$$

(b) show that

$$\frac{dV}{dA}=\frac{(3+\sqrt{5})x}{8\sqrt{3}}$$

The surface area of the icosahedron is increasing at a constant rate of $0.025$ cm$^2$s$^{-1}$

(c) Find the rate of change of the volume of the icosahedron when $x=2$, giving your answer to 2 significant figures.