Math

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Ignoring the walls’ thickness, determine the outside dimensions that will minimize a closed box’s cost if it has a square base and top, if its volume is 32 cubic meters, and if the cost per square meter for the top, bottom, and sides respectively are $.03, $.04, and $.05.

(solve step by step please!)

  • Math -

    4/2

  • Math -

    can you explain how you got that please?

  • Math -

    let the base be x m by x m, let the height be y m
    x^2 y = 32
    y = 32/x^2

    cost = cost of base + cost of top + cost of 4 sides
    = .03x^2 + .04x^2 +4(.05)xy
    = .07x^2 + .2x(32/x^2)
    = .07x^2 + 6.4/x
    d(cost)/dx = .14x - 6.4/x^2
    = 0 for a min of cost
    6.4/x^2 = .14x
    x^3 = .896
    x = .896^(1/3) = appr .964 m
    then y = 32/.964^2 = appr 34.43 m

    The answer seems unreasonable, but I just can't find any errors in my solution.
    Even wrote it out on paper.

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