A large beer keg of height H and cross-sectional area A1 is filled with beer. The top is open to atmospheric pressure. At the bottom is a spigot opening of area A2, which is much smaller than A1.

(a) Show that when the height of the beer is h, the speed of the beer leaving the spigot is approximately.

(b) Show that for the approximation A2 << A1, the rate of change of the height h of the beer is given by.

(c) Find h as a function of time if h = H at t = 0.

(d) Find the total time needed to drain the keg if H = 2 m, A1 = 0.8 m2, and A2 = (10-4) A1.