Simplify formula for rate of change of Volume

**trojsi** · June 17th 2012, 02:28 PM

I need to simplify further the formula for the volume of a cylinder in pdf attached. I also need to interpret it.
Can you give me some hints? thanks

**Prove It** · June 17th 2012, 04:53 PM

Your PDF does not make much sense. I suggest you post the WHOLE question with ALL relevant information, not just the bits you think are appropriate.

**HallsofIvy** · June 17th 2012, 05:05 PM

You are given that the volume of cylinder of radius r and height h is $\pi r^2h$ , that $h= \frac{10}{\pi r^2}$ and that $r= 3+ 2sin(t)$ .

ONE way to do this is to replace h in the volume formula with that formula in terms of r:
$V= \pi r^2h= \pi r^2\frac{10}{\pi r^2}= 10$ and then replace "r" in that by its formula in terms of t.

Oh, wait- there is no "r" in that formula- the " $\pi r^2$ " terms canceled out and we got V= 10, a constant! Well, what does that tell you about its derivative?

**Prove It** · June 17th 2012, 05:10 PM

Originally Posted by HallsofIvy

You are given that the volume of cylinder of radius r and height h is $\pi r^2h$ , that $h= \frac{10}{\pi r^2}$ and that $r= 3+ 2sin(t)$ .

ONE way to do this is to replace h in the volume formula with that formula in terms of r:
$V= \pi r^2h= \pi r^2\frac{10}{\pi r^2}= 10$ and then replace "r" in that by its formula in terms of t.

Oh, wait- there is no "r" in that formula- the " $\pi r^2$ " terms canceled out and we got V= 10, a constant! Well, what does that tell you about its derivative?

This is exactly why I have a feeling that the OP did not post all the information. E.g. looking at the given $\displaystyle \begin{align*} h = \frac{10}{\pi r^2} \end{align*}$ , that implies straight away that the volume is always $\displaystyle \begin{align*} 10 \end{align*}$ , where in the given context it might only start out being $\displaystyle \begin{align*} 10 \end{align*}$ ...

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