Mathematics

Problem

A solid sphere of ice melts uniformly so that its volume is decreasing at a constant rate of

150\,\text{cm}^3

per minute. At the instant when the radius of the sphere is

10\,\text{cm}

, the sphere still retains its shape. (a) Derive an expression for the instantaneous rate of change of the surface area

S

of the sphere in terms of its radius

r

and the instantaneous rate of change of the radius

\frac{dr}{dt}

. (No numerical substitution should appear in this part.) (b) Using the result from part (a) together with the given melting rate, compute

\frac{dS}{dt}

(including units) at the instant when

r = 10\,\text{cm}

. (c) Interpret the sign of

\frac{dS}{dt}

in the physical context of the melting sphere.