Water steadily enters a tank of diameter D _T at a flow rate Q. A small hole in the bottom of the tank (with diameter D ₀ ) allows water to escape. We will show in class that the water velocity through the hole is given by?2gH , where H is the depth of water in the tank.

(a) If the tank is initially empty, derive an expression for the maximum height (H _max ) that the water will reach in the tank. Check your answer using your intuition: is H _max proportional to, inversely proportional to, or independent of D _T , D ₀ and Q? Does this make sense?

(b) Once this maximum height is reached, the flow coming into the tank (i.e. Q) is turned off. If the hole diameter is one-fifth of the tank diameter, and the maximum height is 50 cm, how quickly will the water surface fall as soon as the incoming flow is turned off?

z DT