Question 201030
An open box is to be constructed from a piece of cardboard that is 30in. by 30in. by cutting a square out of each corner and folding up the sides. What are the dimensions of the box that will yield the maximum volume? 
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Draw the 30 by 30 piece.
Draw a square in each of the four corners.
Let the dimensions of each square be "x" by "x".
Imagine cutting out those four squares and folding
up the remaining flaps to make a box with no top.
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The base is "30-2x" by "30-2x" and the height is "x".
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The volume is as follows:
V = x(30-2x)^2
V(x) = 900x - 120x^2 + 4x^3
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Take the derivative and set it equal to zero:
V'(x) = 900 - 240x + 12x^2 = 0
Sove for "x":
2x^2 - 40x + 150 = 0
(x-15)(2x-10)= 0
x = 15 or x = 5
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x = 15 gives you a minumum volume of zero.
x = 5 gives you a maximum volume of 5*20 = 100
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Cheers,
Stan H.