Question 202324
Thomas is going to make an open-top box
by cutting equal squares from the four corners of an
11 inch by 14 inch sheet of cardboard and folding up the
sides. If the area of the base is to be 80 square inches, then
what size square should be cut from each corner?
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Draw a picture of the 80 sq. in sqare.
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Cut an x by x square out of each corner of the cardboard
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Since the cardboard was a square with area 80 sq. in, 
each side was sqrt(80) = 4sqrt(5)
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Fold the sides up to form a height of "x" for the box.
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The base of the box is a square with side = 4sqrt(5)-2x
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The Volume of the box = x[4sqrt(5)-2x] = -2x^2 +4sqrt(5)x
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This is a quadratic with a = -2, b = 4sqrt(5)
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The maximum volume occurs when x = -b/2a = -4sqrt(5)/-4 = sqrt(5)
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So the square you cut out of each corner should have a side of 
length sqrt(5).
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Cheers,
Stan H.