Leaking Cylindrical Tank—Continued When friction and contraction of the water at the...

Leaking Cylindrical Tank—Continued When friction and contraction of the water at the hole are taken into account, the model in Problem 11 becomes

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.

Reference: Problem 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

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 inch. If the tank is initially full, how long will it take to empty?

Problem 13 in Exercises 1.3.

Suppose water is leaking from a tank through a circular hole of area Ah at its bottom. When water leaks through a hole, friction and contraction of the stream near the hole reduce the volume of water leaving the tank per second to

where c (0 < c < 1) is an empirical constant. Determine a differential equation for the height h of water at time t for the cubical tank shown in Figure 1.3.11. The radius of the hole is 2 in., and g = 32 ft/s².