Problem

Figure 4 shows a cylindrical tank that contains some water.

The tank has an internal diameter of $8$ m and an internal height of $4.2$ m.

Water is flowing into the tank at a constant rate of $(0.6\pi)$ m$^3$ per minute.

There is a tap at point $T$ at the bottom of the tank.

At time $t$ minutes after the tap has been opened,

• the depth of the water is $h$ metres
• the water is leaving the tank at a rate of $(0.15\pi h)$ m$^3$ per minute

(a) Show that

$$\frac{dh}{dt}=\frac{12-3h}{320}$$

Given that the depth of the water in the tank is $0.5$ m when the tap is opened,

(b) find the time taken for the depth of water in the tank to reach $3.5$ m.