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Math Help - Maxima/Minima

  1. #1
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    Maxima/Minima

    A can of beans has internal radius r and height h. Given that Surface area is S = 2 pi r ( r + h ) and this is equal to 120 pi. Determine r and h such that the volume of beans in the can is a maximum.
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  2. #2
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    Hello, Jen1603!

    A can of beans has internal radius r and height h.
    Given that surface area is S \:=\:2\pi r(r + h) and this is equal to 120\pi,
    determine r and h such that the volume of the can is a maximum.
    We are given: . 2\pi(r^2+rh) \:=\:120\pi \quad\Rightarrow\quad h \:=\:\frac{60-r^2}{r} .[1]

    The volume of a cylindrical can is: . V \:=\:\pi r^2h .[2]

    Substitute [1] into [2]: . V \;=\;\pi r^2\left(\frac{60-r^2}{r}\right) \quad\Rightarrow\quad V \;=\;\pi(60r - r^3)

    Differentiate and equate to zero: . V' \;=\;\pi(60-3r^2) \:=\:0

    . . and we have: . 3r^2\:=\:60\quad\Rightarrow\quad r^2 \:=\:20 \quad\Rightarrow\quad\boxed{ r \:=\:2\sqrt{5}}

    Substitute into [1]: . h \;=\;\frac{60-20}{2\sqrt{5}} \quad\Rightarrow\quad\boxed{ h \:=\:4\sqrt{5}}

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