A 27 by 27 square piece of cardboard is to be made into a box by cutting out equal square corners from each side of the square. What size corners should be cut out so that the volume of the box is maximized?

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The square piece of cardboard with dimensions 27 by 27 is to be made into an open box by cutting equal squares from the corners and folding the rest appropriately.

Let the length of the sides of the squares cut be x. When the squares are cut and the cardboard folded, the length and with of the box is (27 - 2x) and the height is x. The volume of the box is V = `(27 - 2x)^2*x`

To maximize volume solve V' = 0 for x.

V' = 0

=> `(27 - 2x)^2 + x*2*(-2)*(27 - 2x) = 0`

=> 27 - 2x - 4x = 0

=> 6x = 27

=> x = 27/6 = 4.5

Each side of the corners that are cut should be 4.5 units long.

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