A conical (cone shaped) water tower has a height of 12 ft and a radius of 3 ft. Water is pumped into the tank at a rate of 4 ft^3/min. How fast is the water level rising when the water level is 6 ft?

The water rises at a rate of 16 / (9π) ft/min or approximately 0.566 ft/min.

Step-by-step explanation:

Volume of a cone is:

V = ⅓ π r² h

Using similar triangles, we can relate the radius and height of the water to the radius and height of the tank.

r / h = R / H

r / h = 3 / 12

r = ¼ h

Substitute:

V = ⅓ π (¼ h) ² h

V = ¹/₄₈ π h³

Take derivative with respect to time:

dV/dt = ¹/₁₆ π h² dh/dt

Plug in values:

4 = ¹/₁₆ π (6) ² dh/dt

dh/dt = 16 / (9π)

dh/dt ≈ 0.566

The water rises at a rate of 16 / (9π) ft/min or approximately 0.566 ft/min.