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Frosty the snowman is made from two uniform spherical snowballs, of radii 2R and 3R. The smaller (which is his head) stands on top the larger. As each snowball melts, its volume decreases at a rate which is directly proportional to its surface area. The constant of proportionality being the same for each snowball. During melting, the snowballs remain spherical and uniform.

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When frosty is half his initial height, show that the ratio of his volume to his initial volume is 37 : 224. Let V and h denote Frosty's total volume and height, respectively, at time t. Show that, for 2R $<$ h $\le$ 10R:

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\Large
$\frac{dV}{dh} = \frac{\pi}{8}$
\normalsize
$(h^{2} + 4R^{2})$

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And derive the corresponding expression for 0 $\le$ h $<$ 2R. Sketch \Large $\frac{dV}{dh}$ \normalsize as a function of h, for 4R $\ge$ h $\ge$ 0, hence give a rough sketch of V as a function of h.

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