Posted by andy on Thursday, July 19, 2012 at 9:13pm.
Are you allowed to use differential calculus?
Write an expression for box volume V in terms of the length of the corner cutouts, x.
V = (5-2x)(8-2x)*x
= 4x^3 -26x^2 +40x
Solve for x where V'(x) = dV/dx = 0
dV/dx = 12 x^2 -52x + 40 = 0
3x^2 -13x +10 = 0
x = (1/6)[13 +/- 7] = 1 and 3.333
The second root is not possible, because the removed material would exceed the length of the 5' side.
The largest volume is what you get with x = 1 foot.
Vmax = 18 ft^3
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