Suppose you have a water tank shaped like an inverted cone (tip points down). The height of the cone...?

Question

is 8 meters and the base radius is 8 meters. There is a small hole in the bottom through which water leaks out at a rate proportional to the height of water in the tank. Set up a differential equation that deskcribes the rate of change of the height of water in the tank....

The answer is dh/dt = k/h

why?

Answer 1

At any time the volume of in the cone is (1/3) pi *r*2*h
By similar triangles at any time the radius the top of the water makes is equal to the height
r=h

V = 1/3 pi*h^3

To find dh/dt we can differentiate and solve for dh/dt
Dv/dt = pi*h^2 dh/dt

Solving for dh/dt
dh/dt = [dv/dt] / [pi *h^2]
but dv/dt = kh

dh/dt = [kh] / [pi*h^2]
k/pi is just another constant, and one of the h's cancel

dh/dt = k/h

Answer 2

When we say proportional. we must decide if it is directly proportional or inversely proportional. So the question is should the equation be dh/dt=kh or dh/dt = k/h?

Since the pressure is greater when h is large and less when h is small, the amount of water going through the small hole will be greater when h is large and less when h is small. In other words, more water goes out the hole when h is large and less when h is small. So this is an inverse relationship and the equation is dh/dt=k/h.