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Math Help - Optimization

  1. #1
    Junior Member
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    Optimization

    A solid is formed by adjoining two hemispheres to the ends of a right circular cylinder. The total volume of the solid is 12 cubic centimeters. Find the radius of the cylinder that produces the minimum surface area.

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  2. #2
    Member Henderson's Avatar
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    Since [tex]V = \frac {4}{3} \pi r^3 + \pi r^2 h[/maTH],

    [tex]12 = \frac {4}{3} \pi r^3 + \pi r^2 h[/maTH] and
    h = \frac {12}{\frac {4}{3} \pi r^3 + \pi r^2}

    Also, SA = 4 \pi r^2 + 2 \pi rh, which is

    SA = 4 \pi r^2 + 2 \pi r \frac {12}{\frac {4}{3} \pi r^3 + \pi r^2}.

    Simplify, derive, set equal to zero, and solve.
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