A box with no top of volume 1000 cubic inches is to be made from a 22 by 30 inch piece of cardboard by cutting squares of equal size from the corners, and folding the flaps up. If the box must be at least 4 inches high, what size squares must be cut from each corner?

Question

btaylor · Answer

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If you cut off the corners, you can determine the volume by:
$$V = L \times W \times H = (30-2x)(22-2x)(x)$$$$\rightarrow 1000 = 4x^3 - 104 x^2 + 660$$Make it so it equals zero (by subtracting 1000 from each side), and you have an equation:$$0 = 4x^3 -104x^2+660x-1000$$If you were to graph this function, you could find x by calculating its x-intercepts (this method is a lot easier than trying to group and solve the polynomial).
If you graph it, you get: x = 2.234 and x = 6.47
So, since the problem specifies that the height must be more than 4 inches, x cannot be 2.234. So x = 6.47, and each square is 6.47 inches on each side (41.86 square inches in area)