Question

Suppose a conical tank (think of an ice cream cone, point down) has a capacity of 7 gallons, and that 3 gallons of water are in it already. Water is added at a rate of 6 gallons per minute, but the cone springs a leak near its tip after 10 seconds. Water exits the tank at a rate determined by Torricelli's Law, v2gh (where v is the linear velocity of the water, g the acceleration due to gravity, and h the heigh of the fluid above the hole. If the water is left running, and the hole doesn't grow any larger, will the tank overflow, drain entirely, or will the water reach an equilibrium height?

Answer 1

Suppose at an instant water is filled up to height h. The differential equation governing the height h is by mass balance (in this case volume balance because density is constant):-

where is the vertex angle of the cone, R is the constant flowrate of water added to tank and a is the area of the leak. Simplifying:-

Note that I have assumed t=0 instant corresponding to when the leaking starts. So the water added by the first 10 sec is

which make initial volume = 4 gallon. This implies that h_0 is given by

To do any further analysis, the area of leak is required and the angle theta as well

Although it is more probable that the tank is going to over flow because the leak rate is likely to be very small.