題目

2. (10 points) Consider a circular conic tank shown below. The water is drained at the bottom at the rate of

2\,m^3

per second. Find the rate of change of height

h

when the top circular surface of water has radius

1m

Recall that the volume of a circular cone is $\dfrac{1}{3}\pi r^2 h$ .

解答

由相似形 $r=h$ 。故 $V=\frac{\pi}{3}h^3$ ，對時間微分：

\frac{\mathrm{d}V}{\mathrm{d}t}=\pi h^2\frac{\mathrm{d}h}{\mathrm{d}t}.

水面半徑 $1$ 時 $h=1$ ，又 $\frac{\mathrm{d}V}{\mathrm{d}t}=-2$ ，所以

\boxed{\frac{\mathrm{d}h}{\mathrm{d}t}=-\frac2\pi\text{ m/s}}.