Question

Sand is pouring from a pipe at the rate of 12 cm³/s. The falling sand forms a cone on the ground in such a way that the height of the cone is always one-sixth of the radius of the base. How fast is the height of the sand cone increasing when the height is 4 cm?

Answer 1

The volume of a cone (V) with radius (r) and height (h) is given by,

It is given that,

The rate of change of volume with respect to time (t) is given by,

[By chain rule]

It is also given that.

Therefore, when h = 4 cm, we have:

Hence, when the height of the sand cone is 4 cm, its height is increasing at the rate of.