sand is falling at the rate 27 cubic feet per minute onto a conical pile whose radius is always equal to its height. how fast is the height of the pile growing when the height is exactly (a) 3 feet (b) 6 feet (c) 9 feet.

Step-by-step explanation:

The formula for the volume of a cone is V = (1/3) (area of base) (height). If the radius is always equal to the height of the cone, then V = (1/3) (πh²) (h), where we have eliminated r. Shortened, this comes out to V = (1/3) (π) (h³).

We want to know how fast h is increasing when h = 3 ft.

Taking the derivative dV/dt, we get dV/dt = (1/3) π (3h²) (dh/dt), or, in simpler terms, dV/dt = πh² (dh/dt). Set this derivative = to 27 ft³/min and set h = 3 ft.

Then 27 ft³/min = π (3 ft) ² (dh/dt) and solve for dh/dt: (3/π) ft/min = dh/dt when h = 3 ft.