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Thread: Volume of Metal Block

  1. #1
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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
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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 $\displaystyle 2x^2$ (each end face) $\displaystyle + 4xy$ (each length face).

    $\displaystyle 2x^2 + 4xy = 625$

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

    Now,

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

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

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

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

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

    $\displaystyle 6x^2 = 625$

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

    Once you have calculated this value, substitute into the equation for y. Use the Volume = $\displaystyle x^2y$ formula to find the max volume.
    Last edited by nzmathman; Dec 18th 2008 at 08:14 PM.
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