Model a water tank by a cone 4040 ft high with a circular base of radius 2020 feet at the top. Water is flowing into the tank at a constant rate of 8080 cubic ft/min. How fast is the water level rising when the water is 1212 ft deep

dh / dt =.13 ft / s

Explanation:

Let at height h, radius be r within the cone.

20 / 40 = r / (40-h)

2r = 40 - h

r = (40 - h) / 2

Volume of upper cone

V = 1/3 πr² (40 - h)

= 1/3 π (40 - h) ² x (40 - h) / 4

= 1/12 x π (40 - h) ³

Differentiating on both sides

dV / dt = 1/12 x3 x π (40 - h) ² dh / dt

= π (40 - h) ² / 4 dh / dt

Given

dV / dt = 80, h = 12

80 = π / 4 x (40 - 12) ² dh / dt

dh / dt =.13 ft / s