OK, I'm teaching myself calculus and the book I'm using wasn't particularly clear on this, so I'll give you the problem, the solution, and the part I don't comprehend...

[b]A sponge is in the shape of a right circular cone. As it soaks up the water, it grows in size. At a certain moment, the height equals 6 inches, and is increasing at the rate of .3 inches per second. At that same moment, the radius is 4 inches, and is increasing at the rate of .2 inches per second. How is the volume changing at that time?

So, height = 6 in and is increasing by .3 in/sec

Radius is 4 inches and is increasing by .2 in/sec

The volume of a right circular cone is $\displaystyle V = \frac{1}{3}[pi]r^2h$ (If someone could tell me how to write the symbol for pi in the [ Tex ] tags, that'd be helpful in the future.)

Now is where I get confused, to derive that formulain respect to the variable twhich is time. As well, I was unsure of how to apply the product rule when I have three different numbers being multiplied at once...

The book gives this;

$\displaystyle \frac{dV}{dt} = \frac{1}{3}[pi][2r\frac{dr}{dt}h + r^2\frac{dh}{dt}]$

I realize that $\displaystyle \frac{dr}{dt}$ is the rate the radius is increasing and $\displaystyle \frac{dh}{dt}$ is the rate the height is increasing, but knowing when/ where/ why to use them is what I don't get.

I think someone may just need to explain each step in the problem and I'll get it.