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Math Help - Designing a cyclinder with minimal surface area

  1. #1
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    Designing a cyclinder with minimal surface area

    You are designing a new cylindrical can to hold 1 liter (1000 cubic centimeters) of Jake's Special Paint. Find the dimensions that will minimize the cost of metal to manufacture the can.
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  2. #2
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    V={\pi}r^{2}h=1000..........[1]

    S=2{\pi}rh+2{\pi}r^{2}..........[2]


    Solve [1] for, say, h and sub into [2]:

    h=\frac{1000}{{\pi}r^{2}}

    S=2{\pi}r(\frac{1000}{{\pi}rh})+2{\pi}r^{2}=2{\pi}  r^{2}+\frac{2000}{r}

    This is what is to be minimized. Differentiate, set to 0 and solve for r. Then h will follow.

    I believe you will find that the minimum is achieved when the diameter equals the height.
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  3. #3
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    is the derivative:


    4 * pi * r - 2000 * r ^ -2
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  4. #4
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    or simplified...

    4*pi*r-(2000/x^2)
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