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Math Help - Optimization

  1. #1
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    Optimization

    A closed box with a square base has a volume of 250 cubic meters. The material for the top and bottom of the box cost $2 per square meter, and the material for the sides cost $1 per square meter. Can the box be constructed for less then $3000?
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  2. #2
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    Quote Originally Posted by lisa1984wilson View Post
    A closed box with a square base has a volume of 250 cubic meters. The material for the top and bottom of the box cost $2 per square meter, and the material for the sides cost $1 per square meter. Can the box be constructed for less then $3000?

    Let the side of the base be of length x and the height y.

    The volume is then x^2y = 250.

    Top + Bottom = x^2 + x^2 = 2x^2 square meters, at 2 dollars each, this gives a top+bottom cost of 4x^2.

    Sides' area = 4xy, at one dollar per square meter, this gives a cost of 4xy dollars

    Total Cost = 4xy + 4x^2
    Restriction: x^2 y= 250
    Restriction: Total Cost \leq 3000

    Total cost = 4x \frac{250}{x^2} + 4x^2 = \frac{1000}{x} + 4x^2

    Optimal cost:

    C'(x) = -1000/x^2 + 8x = 0

    8x^3 - 1000 = 0
    x = 5
    C(5) = 200 + 100 = 300

    C(5) is a minimum because C"(x) = \frac{2000}{x^3} + 8 > 0 at x=5.

    The minimum cost is $300. So then your answer would be no, you cannot build this box with less than 300 dollars, but you can build it with exactly 300 dollars.
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