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Math Help - Simplify formula for rate of change of Volume

  1. #1
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    Simplify formula for rate of change of Volume

    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
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  2. #2
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    Re: Simplify formula for rate of change of Volume

    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.
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    Re: Simplify formula for rate of change of Volume

    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?
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    Re: Simplify formula for rate of change of Volume

    Quote Originally Posted by HallsofIvy View Post
    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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