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Q: You cut square corners with side lengths that are whole numbers from a piece of cardboard with dimensions 20 inches by 30 inches. You then fold the cardboard to create a box with no lid. Which of the
following dimensions will give you the greatest volume? A. 12 in. by 22 in. by 4 in. B. 10 in. by 20 in. by 5 in. C. 14 in. by 24 in. by 2 in. D. 10 in. by 24 in. by 6 in
A: You cut square corners with side lengths that are whole numbers from a piece of cardboard with dimensions 20 inches by 30 inches. You then fold the cardboard to create a box with no lid. 12 in. by 22 in. [ by 4 in will give you the greatest volume. ***Length of box base: 20 - 2x Width of box base: 30 - 2x Height of box: x Volume of box: V = x(20 - 2x)(30 - 2x) V = x(600 - 40x - 60x +
4x^2) V = x(600 - 100x + 4x^2) V = 600x - 100x^2 + 4x^3 dV/dx = 600 - 200x + 12x^2 Set that to zero: 12x^2 - 200x + 600 = 0 x = (-(-200) +/- sqrt((-200)^2 - 4(12)(600))) / (2*12) x = (200 +/- sqrt(40000 - 28800)) / 24 x = (200 +/- sqrt(11200)) / 24 x = (200 +/- sqrt(11200)) / 24 x =~ 12.74 or 3.92 Since you want whole numbers, x = 13 or 4 If x = 13, V = -312, but a negative number doesn't make sense, so we'll ignore that If x = 4, V = 1056, so that's the maximum volume The dimensions are: Length of box base: 12 Width of box base: 22 Height of box: 4 ]
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