A conical water tank with vertex down has a radius of 12 feet at the top and is 26 feet high. If water flows into the tank at a rate of 30 {\rm ft}^3{\rm /min}, how fast is the depth of the water increasing when the water is 12 feet deep?

Answer 1

Make your sketch,
let the radius of the water level be r ft
let the height of the water be h ft

by ratios:
r/h = 12/26
26r = 12h
r = 6h/13

Vol = (1/3)π r^2 h
= (1/3)π (36h^2/169) g
= (12/169)π h^3
d(Vol)/dt = (36/169)π h^2 dh/dt

30 = (36/169)π (144) dh/dt
dh/dt = 30(169/(36π(144))
= 845/(864π) ft/min = appr .3113 ft/min

check my arithmetic, I should have written it down instead of doing on the screen only