Question

sand falling onto a conical pile at a rate of 10 cubic feet/min. diameter of the base of the cone is 3times the altitude. at what rate is the height of the pile changing when the pile is 15ft high?

Answer 1

we know
V= pi/3 r^2 *h also given 2r =3h hence r =3h/2 (assume r and h to be fucnction of time t and hence V is also function of t)
thus we have
V= pi/3 *9*h^2/4 *h
or
V=3pi/4 *h^3
now diiferentiating both side wrt t
we have
dV/dt =9 pi/4 * h^2 dh/dt
here given dV/dt = 10 cubic feet/min and h =15 we need to find dh/dt