Question

Ement is poured in a construction site at a rate of 3 m3/min, and its composi- tion is such that it forms a pile in the shape of a cone. the diameter and height of the cone are always equal. the volume of the cone is given by v = π 12d2h , with d the diameter and h the height. how fast is the height of the pile increasing when the pile is 3m high?

Answer 1

The height of the pile is increasing at a rate of 1/π units per minute.

Step-by-step explanation:

To find the rate at which the height of the pile is increasing, we need to find the derivative of the volume equation with respect to time, and then substitute the given values. The volume of the cone can be expressed as V = π/12 * d^2 * h, where d is the diameter and h is the height. Taking the derivative of this equation with respect to time gives us dV/dt = π/3 * d * dh/dt. Given that d = h, we can substitute for d in the derivative equation to get dV/dt = π/3 * h * dh/dt.

Given that the volume rate of change is 3 m^3/min, we can substitute dV/dt = 3 and solve for dh/dt:

3 = π/3 * h * dh/dt

dh/dt = 3 / (π/3 * h)

Now, substituting h = 3 m, we can find the rate at which the height of the pile is increasing:

dh/dt = 3 / (π/3 * 3) = 1 / π