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?

Let V be the volume of the cone, then

It is given that, h=

⇒ r = 6h

Therefore,

Now, the rate of change of its volume with respect to time (t) is given by:

Now, the rate of change of its volume with respect to x is as

……… by chain rule

…………….(1)

Now, it is also given that and h = 4cm

So, putting the value in equation (1), we get

Therefore, the height of the sand cone increasing when the height is 4 cm is cm/s.