Find the maximum volume of a box that can be made by cutting out squares from the corners of an 8-inch by 15-inch rectangular sheet of cardboard and folding up the sides.

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Mathematics

Azaria Hampton

12 May, 01:09

Find the maximum volume of a box that can be made by cutting out squares from the corners of an 8-inch by 15-inch rectangular sheet of cardboard and folding up the sides.

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Leo Hunter · Answer 1

Answers: V = (15-2x) (8-2x) x V = 120x - 46xÂ˛ + 4xÂł V' = 120 - 92x + 12xÂ˛ 0 = 120 - 92x + 12xÂ˛ x = (92Â±52) / 24 x = 6 or 5/3 Pick x = 5/3 Volume = (15-2*5/3) (8-2*5/3) * 5/3 = 90.74 inÂł To justify x = 5/3 is the max V" = - 92 + 24x V" (5/3) = - 92 + 24 (5/3) <0 negative. Volume V (5/3) is max because the function is concave down with V"<0.