4.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{fe07afad-9cfc-48c0-84f1-5717f81977d4-10_378_332_246_808}
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\caption{Figure 1}
\end{figure}
A regular icosahedron of side length \(x \mathrm {~cm}\), shown in Figure 1, is expanding uniformly. The icosahedron consists of 20 congruent equilateral triangular faces of side length \(x \mathrm {~cm}\).
- Show that the surface area, \(A \mathrm {~cm} ^ { 2 }\), of the icosahedron is given by
$$A = 5 \sqrt { 3 } x ^ { 2 }$$
Given that the volume, \(V \mathrm {~cm} ^ { 3 }\), of the icosahedron is given by
$$V = \frac { 5 } { 12 } ( 3 + \sqrt { 5 } ) x ^ { 3 }$$
- show that \(\frac { \mathrm { d } V } { \mathrm {~d} A } = \frac { ( 3 + \sqrt { 5 } ) x } { 8 \sqrt { 3 } }\)
The surface area of the icosahedron is increasing at a constant rate of \(0.025 \mathrm {~cm} ^ { 2 } \mathrm {~s} ^ { - 1 }\)
- Find the rate of change of the volume of the icosahedron when \(x = 2\), giving your answer to 2 significant figures.