A stalactite has the shape of a circular cone. Its height is 200 mm and is increasing at a rate of 3 mm per century. Its base radius is 40 mm and is decreasing at a rate of 0.5 mm per century. Determine if its volume is increasing or decreasing, and the rate at which the volume is changing.

\(V = \frac{\pi }{3}{r^2}h\)

\(\frac{{{\text{d}}V}}{{{\text{d}}t}} = \frac{\pi }{3}\left[ {2rh\frac{{{\text{d}}r}}{{{\text{d}}t}} + {r^2}\frac{{{\text{d}}h}}{{{\text{d}}t}}} \right]\)     M1A1A1

at the given instant

\(\frac{{{\text{d}}V}}{{{\text{d}}t}} = \frac{\pi }{3}\left[ {2(4)(200)\left( { - \frac{1}{2}} \right) + {{40}^2}(3)} \right]\)     M1

\( = \frac{{ - 3200\pi }}{3} = - 3351.03 \ldots  \approx 3350\)     A1

hence, the volume is decreasing (at approximately 3350 \({\text{m}}{{\text{m}}^3}\) per century)     R1

[6 marks]

Few candidates applied the method of implicit differentiation and related rates correctly. Some candidates incorrectly interpreted this question as one of constant linear rates.