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Optimization problem?
Square corners are to be cut out of a rectangular 12 by 24 inch piece of cardboard to form an open topped box. What size should the cut squares be to maximize the volume of the box? (you do not need to calculate the maximal volume, just the size of the square.)
1 Answer
- Steve ALv 76 years ago
square is x by x
Length = 24 - 2x
width = 12 - 2x
height = x
volume = (24-2x)(12-2x)x
V = 4x^3 - 72x^2 + 288x
V' = 12x^2 - 144x + 288 = 0
x^2 - 12x + 24 = 0
(x-6)^2 = 12
x = 6 +/- sqrt(12)
x = 2.54 ; x = 9.46
If you cut 9.46 from each side, the 12" side will be gone.
Answer x = 6 - 2sqrt(3) or about 2.54"
