Chapter 3.2, Problem 13E

Leaking Conical Tank A tank in the form of a right-circular cone standing on end, vertex down, is leaking water through a circular hole in its bottom.

1. (a) Suppose the tank is 20 feet high and has radius 8 feet and the circular hole has radius 2 inches. In Problem 14 in Exercises 1.3 you were asked to show that the differential equation governing the height h of water leaking from a tank is

$\frac{dh}{dt}=\frac{5}{6h^{3/2}}.$

In this model, friction and contraction of the water at the hole were taken into account with c = 0.6, and g was taken to be 32 ft/s². See Figure 1.3.12. If the tank is initially full, how long will it take the tank to empty?

1. (b) Suppose the tank has a verte x angle of 60° and the circular hole has radius 2 inches. Determine the differential equation governing the height h of water. Use c = 0.6 and g = 32 ft/s². If the height of the water is initially 9 feet, how long will it take the tank to empty?