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Math Help - Optimization: Max Capacity of Cone-Shaped Cup

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    Optimization: Max Capacity of Cone-Shaped Cup

    A cone shaped 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.
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  2. #2
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    Re: Optimization: Max Capacity of Cone-Shaped Cup

    R = sector radius (a given constant)

    r = cone radius

    h = cone height


    R^2 = r^2 + h^2

    r^2 = R^2 - h^2

    V = \frac{\pi}{3} r^2 h

    V = \frac{\pi}{3} (R^2 - h^2)h

    V = \frac{\pi}{3} (R^2 h - h^3)

    \frac{dV}{dh} = \frac{\pi}{3} (R^2 - 3h^2)

    \frac{dV}{dh} = 0 and is a maximum when h = \frac{R}{\sqrt{3}} since \frac{d^2V}{dh^2} < 0 for all h

    V_{max} = \frac{\pi}{3} \left(R^2 - \frac{R^2}{3}\right) \frac{R}{\sqrt{3}}

    V_{max} = \frac{2 \pi R^3}{9\sqrt{3}}
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