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4 August, 21:15

The volume V of a right circular cylinder of height 3 feet and radius r feet is V=V (r) = 3 (pi) (r^2). find the instantaneous rate of change of the volume with respect to the radius at r=3.

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  1. 4 August, 21:26
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    When I see the words "instantaneous rate of change", I have to assume that you're in some stage of pre-calculus in your math class.

    The instantaneous rate of change of a function is just its first derivative.

    We have the function

    V (r) = 3 π r²

    and we need its first derivative with respect to ' r '. That shouldn't be

    too hard, because the ' 3 π ' is nothing but constants.

    Watch me while I do it slowly for you:

    - - The derivative of ' r² ' with respect to ' r ' is ' 2r '.

    - - The derivative of V (r) with respect to ' r ' is (3 π) times the derivative of ' r² '.

    - - The derivative of V (r) with respect to ' r ' is (3 π) times (2r) = 6 π r.

    The value of the derivative when r=3 is (6 π 3) = 18π = about 56.5 feet³/foot.
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