A tank in the form of a right circular cylinder standing on end is leaking water through a circular hole in its bottom. As we saw in (10) of Section 1.3, when friction and contraction of water at the hole are ignored, the height h of water in the tank is described by

dh/dt = - A_h/A_w √2gh,

where Aw and Ah are the cross-sectional areas of the water and the hole, respectively.

(a) Solve the DE if the initial height of the water is H. By hand, sketch the graph of h(t) and give its interval I of defi­nition in terms of the symbols A_w, A_h, and H.

Use g = 32 ft/s².

(b) Suppose the tank is 10 feet high and has radius 2 feet and the circular hole has radius ½ inch. If the tank is initially full, how long will it take to empty?