A conical cistern is 10 ft. across the top and 12 ft. deep. If water is poured into the cistern at the rate of 1 cubic foot per second, how fast is the surface rising when the water is 8 ft. deep?

Answer 1

let the water level have a radius of r ft
let the height of the water be h ft
by ratio:
r/h = 5/12
12r = 5h ---> r = 5h/12

V = (1/3)π r^2 h
= (1/3)π(25h^2/144)(h)
= (25/432)π h^3
dV/dt = (25/144)π h^2 dh/dt

plug in our given stuff
1 = (25/144)π(64) dh/dt
dh/dt = 144/(25(64)π) = 9/(100π)

check my arithmetic