# calculus

An open box is to be made out of a 10-inch by 14-inch piece of cardboard by cutting out squares of equal size from the four corners and bending up the sides. Find the dimensions of the resulting box that has the largest volume.

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1. L = length = 14 - 2h
w = width = 10 - 2 h

V = L * w * h
= (14-2h)(10-2h)(h)
= (140 - 28 h - 20 h + 4 h^2 )h
= 4 h^3 -48 h^2 + 140 h
so
dV/dh = 0 at max
0 = 12 h^2 - 96 h + 140
0 = 3 h^2 - 24 h + 35

h = [ 24 +/- sqrt (576-420) ] /6
h = 4 +/- (1/6)(12.5)
h = 6.08 or 1.92
6.08 is impossible so 1.92 is height

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