related rates?

[Yoda]

The volume of a cone is expanding(radius) at a rate of 2cm/sec. The cone has a fixed height of 90cm, and the volume of the cone is given by V=1/3pir^2h. Find the rate of change of volume when the radius is 6cm.

[Stevensmay]

First off identify all the things we are given in the question.



The question asks us to find the rate at which the volume is changing, implying that it has something to do with time. Therefore we can see that the question is asking us to find , given a radius.

But we don't have  , so we need to find a way to get to it.
One way to do this is to rearrange what we have in some way, so that we end up with it.

In this case we can say 
The dr will 'cancel' out, leaving us with what we wanted.

We already know what is from the question, so we just need to find

From the question we have , but we cannot differentiate with respect to two variables. This is easily fixed, as we are told h (height) will always be 90cm, so we can just substitute it in.





Differentiate as normal.



So now we have both our derivatives, we can multiply them together as we said earlier to get to





So now we have an equation that relates a given radius, to the rate of change of the volume.

Substituting in the given radius of 6 gives us

Therefore when the radius is 6cm, the rate of change is