Question

Suppose that sand is collecting in the shape of a cone in such a way that the base radius of the cone is always one-third of its height. If 3cm^3/min is the rate at which the sand is being added to the cone, how fast is the height of the cone changing when it is 7cm tall?

Answer 1

v = 1/3 π r^2 h ... r = h / 3

v = 1/3 π (h / 3)^2 h = π/27 h^3

dv/dt = π/9 h^2 dh/dt

3 cm^3/min = π/9 * 49 cm^2 * dh/dt