Calc

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A closed box with a square base is to be constructed so that its volume is 324 cubed feet. The material for the top and bottom cost is $3 per square foot, and that for the sides $2 per square foot. Find the dimensions of the box so that the cost will be minimum.

  • Calc -

    top and bottom are w by w
    height is h

    324 = w^2 h so h = 324/w^2

    Cost = 6 w^2 + 4 w h
    = 6 w^2 + 4 w (324/w^2)
    = 6 w^2 + 1296/w

    dC/dw = 0 for min = 12 w -1296/w^2
    0 = 12 w^3 - 1296
    w^3 = 108
    w = 4.76
    h = 14.3

  • Calc -

    Are you sure about the 4w (324/w^2)?
    Shouldn't it be 8w (324/w^2), since you are multiplying the four sides by the cost of $2 per square foot?

  • Calc -

    Yes, you're right.
    If we proceed with
    Cost = 6 w^2 + 8 w h
    we get:
    w=6 and h=9.

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