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Math Help - Volume and partial derivatives

  1. #1
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    Nov 2008
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    Question Volume and partial derivatives

    How to solve the following problem? How can I start the following problem?

    The base of an aquarium with given volume V is made of slate and the sides are mage of glass. If slate costs five times as much (per unit area) as glass, find the dimensions of the aquarium that minimize the cost of the materials.

    Thanks for any type of help you may provide me!
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  2. #2
    Super Member
    Joined
    Aug 2008
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    Let the slate base be x by y. Let the sides have height z. Then the cost function is:

    C(x,y,z)=5xy+2xz+2yz

    but V=xyz or z=\frac{V}{xy}

    so:

    C(x,y)=5xy+2x\left(\frac{V}{xy}\right)+2y\left(\fr  ac{V}{xy}\right)

    Now start taking partials and then use the Second Derivative test to test for extrema. I did it quick and got x=\sqrt[3]{2/5 V} and y=\sqrt[3]{2/5 V} as the extrema. But double check it.
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