Math

A Candy box is made from a piece of cardboard that meaasures 11 by 7 inches. Squares of equal size will be cut out of each corner. The sides will then be folded up to form a rectangular box. What size square should be cut from each corner to obtain maximum volume?

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  1. Let squares of size x" be cut from the corners.

    Volume of (open) box
    =V(x)
    =height*length*width
    =x(11-2x)(7-2x)
    =4x^3-36x^2+77x

    For maximum (or minimum), equate derivative to zero:
    dV(x)/dx = 12x²-72x+77=0
    Solve for x to get
    x=3±(√93)/6
    =1.39 or 4.61 (approximately)

    4.61 is clearly not a feasible solution (because 2*4.61 > 7") and will be rejected.

    So the cut-outs will be squares of 1.39" (approximately).

    Now verify that the solution so obtained is a maximum by ensuring that d²V(x)/dx² < 0:
    d²V(x)/dx² = 24x-72 = -38.6 <0 OK.

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