Calculus question

Question

Hafsa is bored and is pouring sand onto the floor. The poured-out sand forms a conical pile whose height is three-quarters of its radius and whose height is growing at a rate of 1.5 cm per second. How fast is Hafsa pouring the sand at the instant that the pile of sand is 3 cm high?

Answer 1

The volume of a cone is given by:

V = 1/3πr²h

We also know that h = 3/4r. Given that, we can say that r = 4/3h. Substituting "4/3h" for "r" in the volume equation gives us:

V = 1/3π(4/3h)²(h)

V = 16/27πh³

Taking the derivative with respect to time (and ensuring we use the chain rule), we get:

dV/dt = 16/27π(3h²)(dh/dt)

We are given that h = 3 and that dh/dt = 1.5 so:

dV/dt = 16/27π(3(3)²)(1.5) = 24π cm³/sec (approx 75.4 cm³/sec)