Chapter 3.2, Problem 12E

Leaking cylindrical tank—continued When friction and contraction of the water at the hole are taken into account, the model in Problem 11 becomes

$\frac{dh}{dt}=-c\frac{A_h}{A_w}\sqrt{2gh},$

where 0 < c < 1. How long will it take the tank in Problem 11(b) to empty if c = 0.6? See Problem 13 in Exercises 1.3.

11. Leaking cylindrical tank 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

$\frac{dh}{dt}=-\frac{A_h}{A_w}\sqrt{2gh},$

where A_w and A_h are the cross-sectional areas of the water and the hole, respectively.

(b) Suppose the tank is 10 feet high and has radius 2 feet and the circular hole has radius $\frac{1}{2}$ inch. If the tank is initially full, how long will it take to empty?