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Math Help - Volume of Metal Block

  1. #1
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    Dec 2008
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    Volume of Metal Block

    How do i solve this:

    A rectangular block of metal with a square cross-section has a total surface area of 625cm2. Find the maximum volume of the block of metal?

    Help!!
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  2. #2
    Member
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    Dec 2008
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    Auckland, New Zealand
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    Let the rectangular block have dimensions x, x and y. (2 x's because the cross section is a square).

    So the surface area is given be 2x^2 (each end face) + 4xy (each length face).

    2x^2 + 4xy = 625

    y = \frac{625 - 2x^2}{4x}

    Now,

    Volume = 'area of cross section' times 'length' = x^2y.

    = x^2 \times \frac{625 - 2x^2}{4x}

    = x \times \frac{625 - 2x^2}{4}

    = \frac{1}{4} \left({625x - 2x^3}\right)

    \frac{dV}{dx} = \frac{1}{4} (625 - 6x^2) = 0 (for max)

    6x^2 = 625

    x = \sqrt{\frac{625}{6}}

    Once you have calculated this value, substitute into the equation for y. Use the Volume = x^2y formula to find the max volume.
    Last edited by nzmathman; December 18th 2008 at 09:14 PM.
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