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Math Help - Optimization problem, stuck

  1. #1
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    Optimization problem, stuck

    A conical drinking cup is made from a circular piece of paper of radius R by cutting out a sector and joining the edges CA and CB. Find the maximum capacity of such a cup

    link of picture of the cup is found in the attachement
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  2. #2
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    Hello, aaasssaaa!

    A conical drinking cup is made from a circular piece of paper of radius R
    by cutting out a sector and joining the edges CA and CB.
    Find the maximum capacity of such a cup.
    Code:
                ..*.*.*..
             .*:::::::::::*.
           A*:::::::::::::::*B
           *  *:::::::::::*  *
              R *:::::::* R
          *       *:::*       *
          *         *         *
          *         C         *
    
           *                 *
            *               *
              *           *
                  * * *

    The side view of the cup looks like this:
    Code:
                   r
          *-----+-----*
           \    :    /
            \  h:   /
             \  :  / R
              \ : /
               \:/
                *

    where r is the radius of the cone and h is its height.

    We see that: . r^2 + h^2\;=\;R^2\quad\Rightarrow\quad r^2 \;=\;R^2 - h^2 [1]


    The volume of a cone is: . V \;=\;\frac{\pi}{3}r^2h [2]

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

    Maximize V:\;\;V' \;=\;\frac{\pi}{3}\left(R - 3h^2\right)\;=\;0\quad\Rightarrow\quad h^2 \:=\:\frac{R^2}{3}

    . . and we get: . \boxed{h \:= \:\frac{\sqrt{3}R}{3}}

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

    Substitute into [2]: . V \;=\;\frac{\pi}{3}r^2 h\;=\;\frac{\pi}{3}\left(\frac{2R^2}{3}\right)\lef  t(\frac{\sqrt{3}R}{3}\right)


    Therefore, the maximum volume is: . V \;=\;\boxed{\frac{2\pi\sqrt{3}}{27}R^3}

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  3. #3
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    Hey what program did you use to right your numbers, roots and pie in
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  4. #4
    Grand Panjandrum
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    Quote Originally Posted by aaasssaaa View Post
    Hey what program did you use to right your numbers, roots and pie in
    The mathematical typesetting here is done using a LaTeX see this
    (to see the code generating the equations quote the message and
    you will see the typesetting codes embedded in the message).

    RonL
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