A large industrial hopper in the shape of an inverted square-based pyramid is being filled with fine grain at a constant rate of 24 cm3/s24\text{ cm}^3/\text{s}24 cm3/s.
After ttt seconds, the depth of the grain in the hopper is h cmh\text{ cm}h cm. The apex of the pyramid is at the bottom.
When the depth of the grain is h cmh\text{ cm}h cm, the volume V cm3V\text{ cm}^3V cm3 of the grain is given by
V=29h3 V = \frac{2}{9}h^3 V=92h3Show that when t=3t = 3t=3,
dVdh=12183 \frac{dV}{dh} = 12\sqrt[3]{18} dhdV=12318Hence, find the rate at which the depth of the grain is increasing when t=3t = 3t=3. Give your answer to three significant figures.
