Question

Suppose water is leaking from a tank through a circular hole of area

A_h

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 cA_h

2gh

, where

c (0 < c < 1)

is an empirical constant.

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. (Assume the removed apex of the cone is of negligible height and volume.)

(a)

Suppose the tank is 20 feet high and has radius 8 feet and the circular hole has radius 2 inches. The differential equation governing the height hin feet of water leaking from a tank after t seconds is

dh

dt

= −

5

6h^3/2

. In this model, friction and contraction of the water at the hole are taken into account with

c = 0.6,

and g is taken to be

32 ft/s².

See the figure below.A right-circular conical tank containing water is shown.

• The cone opens upward and the point of the cone at the bottom is labeled: circular hole.
• A dashed line extends horizontally from the circular hole.
• The surface of the water in the tank is labeled: A_w and a line segment from the dashed line to A_w is labeled: h.
• The radius at the top of the tank is labeled: 8 ft.
• The height of the tank is labeled: 20 ft.

Solve the initial value problem that assumes the tank is initially full.

h(t) =  

If the tank is initially full, how long (in minutes) will it take the tank to empty? (Round your answer to two decimal places.)

minutes

(b)

Suppose the tank has a vertex angle of 60° and the circular hole has radius 4 inches. Determine the differential equation governing the height hof water. Use

c = 0.6

and

g = 32 ft/s².

dh

dt

=  

Solve the initial value problem that assumes the height of the water is initially 10 feet.

h(t) =  

If the height of the water is initially 10 feet, how long (in minutes) will it take the tank to empty? (Round your answer to two decimal places.)

minutes

Answer 1