Calculus

An OPEN box has a square base and a volume of 108 cubic inches and is constructed from a tin sheet. Find the dimensions of the box, assuming a minimum amount of material is used in it's construction. HINT: the goal is to minimize the surface are of the OPEN box; this is the function that needs to be minimized. Express the function as a function of one variable using the fact that the volume is 108 cubic inches.

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  1. volume = v = x^2 h
    area = a = x^2 + 4 x h

    h = 108/x^2

    a = x^2 + 4 x (108/x^2)
    a = x^2 + 432/x
    da/dx = 0 = 2 x -432/x^2
    432 = 2 x^3
    x^3 = 216
    x = 6
    h = 108/36 = 3

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