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Math Help - Maximization/minimization/whatever

  1. #1
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    Maximization/minimization/whatever

    "The volume of a cylindrical tin can with a top and a bottom is to be 16pi cubic inches. If a minimum amount of tin is to be used to construct the can, what must be the height, in inches, of the can?"

    I know I need to take the derivative of the volume of a cylinder.
    After that, though, I'm lost.
    Please help.
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  2. #2
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    Quote Originally Posted by MadHotThunder View Post
    "The volume of a cylindrical tin can with a top and a bottom is to be 16pi cubic inches. If a minimum amount of tin is to be used to construct the can, what must be the height, in inches, of the can?"

    I know I need to take the derivative of the volume of a cylinder.
    After that, though, I'm lost.
    Please help.
    Let r be the radius and h the height of the can. The volume is fixed so

    V = \pi r^2 h = 16\pi

    The surface area is

    A = 2 \pi r^2 + 2 \pi r h

    From the volume formula h = \frac{16}{r^2} so the area is

    A = 2 \pi r^2 + \frac{32 \pi}{r}

    Now use calculus to find the r that minimizes A. Once you have that you can use the volume formula to find h.
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  3. #3
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    Quote Originally Posted by danny arrigo View Post
    Let r be the radius and h the height of the can. The volume is fixed so

    V = \pi r^2 h = 16\pi

    The surface area is

    A = 2 \pi r^2 + 2 \pi r h

    From the volume formula h = \frac{16}{r^2} so the area is

    A = 2 \pi r^2 + \frac{32 \pi}{r}

    Now use calculus to find the r that minimizes A. Once you have that you can use the volume formula to find h.
    That's where the problem comes in. I don't know how to do that.
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