Rate of change of h.A right circular cylinder has a diameter of 12 in. and a height of 12 in. If water is flowing in at the rate of 4pi in^3 per minute, find the rate of change of the height when the height is 4 in
So we need to find dh/dt.
At any time t,
Volume, V = (pi r^2)h
Since r does not change with time, then r is a constant, so,
V = pi (12/2)^2 h
V = (36pi)h
Differentiate both sides with respect to time t,
dV/dt = (36pi)(dh/dt) -------------(i)
Since dV/dt is given as 4pi cu.in. per min,
4pi = (36pi)(dh/dt)
dh/dt = 4pi / 36pi
dh/dt = 1/9 in./min. -----------answer.
In (i), we see that the variables are (dV/dt) and (dh/dt) only. That means h is not factor, so, at any h, at h=1, h=4, h=9, the dh/dt is the same until the cylinder is filled up.