Example

A funnel is in the shape of a right circular cone with its vertex pointing downward, its axis being vertical, the cone’s base is of radius 6cm, and the cone’s height is 12cm. A liquid is poured in the funnel at the rate of . In the same time the liquid is leaking from a hole in the vertex at the rate of

Calculate:

1-    The rate at which the liquid level is rising when it is 6 cm deep.

2-    The rate at which the radius of the liquid surface is increasing at this moment.

3-     The time needed to fill the funnel.

Solution

Let the radius of the liquid surface be r cm after t seconds, and let the height of the liquid at this moment be h cm. The volume V of the liquid inside the cone is increasing at a rate of . We know that:

We see that r=0.5h. Therefore,

By differentiating both sides with respect to t:

When h=6 cm, and , we get,

cm/sec

Having h=2r, we can get dh/dt=2dr/dt. Thus,

The radius is increasing at this rate we found.

The tunnel fills when:

As , the time required to fill the funnel = .