A regular icosahedron of side length $$\(x \mathrm{~cm}\)$$, shown in Figure 1, is expanding uniformly. The icosahedron consists of 20 congruent equilateral triangular faces of side length $$\(x \mathrm{~cm}\)$$. (a) Show that the surface area, $$\(A \mathrm{~cm}^{2}\)$$, of the icosahedron is given by $$\[ A=5 \sqrt{3} x^{2} \]$$ (2) Given that the volume, $$\(V \mathrm{~cm}^{3}\)$$, of the icosahedron is given by $$\[ V=\frac{5}{12}(3+\sqrt{5}) x^{3} \]$$ (b) show that $$\(\frac{\mathrm{d} V}{\mathrm{~d} A}=\frac{(3+\sqrt{5}) x}{8 \sqrt{3}}\)$$ (3) The surface area of the icosahedron is increasing at a constant rate of $$\(0.025 \mathrm{~cm}^{2} \mathrm{~s}^{-1}\)$$ (c) Find the rate of change of the volume of the icosahedron when $$\(x=2\)$$, giving your answer to 2 significant figures. (3)

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