SOLUTION: A box with a square bottom and no top is to be made from a 6 by 6 - inch piece of material by cutting equal sized squares from the corners and then turning up the sides. What sho
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Question 1155483: A box with a square bottom and no top is to be made from a 6 by 6 - inch piece of material by cutting equal sized squares from the corners and then turning up the sides. What should the dimensions of the squares be if the box is to have maximum volume? Found 2 solutions by josgarithmetic, ikleyn:Answer by josgarithmetic(39617) (Show Source):
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A box with a square bottom and no top is to be made from a 6 by 6 - inch piece of material by cutting equal sized squares
from the corners and then turning up the sides. What should the dimensions of the squares be if the box is to have maximum volume?
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When you cut these 4 squares and turn up the sides, you will have the base of the box as a square
with the sides of (6-2x) inches in each direction.
So, the area of the bottom to this box will be (6-2x)^2; hence, the volume of the box will be x*(6-2x)^2.
Thus, the volume as a function of x is
V(x) = x*(6-2x)^2 = x*(36 - 24x + 4x^2) = 4x^3 - 24x^2 + 36x.
To find the maximum to this function, differentiate the function and equate the derivative to zero
0 = 12x^2 - 48x + 36.
The right side is factorable
0 = 12*(x^2 - 4x + 3) = 12*(x-3)*(x-1).
So the roots to this equation are x= 3 or x= 1.
Of the two roots, x= 3 is not the solution to the problem, since we then have zero size for the bottom of the box.
Therefore, only x= 1 makes sense.
The volume then is V = 1*(6-2x)^2 = 1*(6 - 2*1)^2 = 1*4^2 = 1*16 = 16 cubic inches.
ANSWER. The size of the squares to cut is 1 inch.
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