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An open box is to be made from a square piece of cardboard, 36 inches by 36 inches, by removing a small square from each corner and folding up the flaps to form the sides. What are the dimensions of the box of greatest volume that can be constructed in this way?

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sqdancefan

9514 1404 393

Answer:

  24 in square by 6 in deep

Step-by-step explanation:

Let x represent the side of the square cut from each corner. Then the dimensions of the base of the box are 36-2x in each direction. The total volume of the box is ...

  V = LWH = (36 -2x)(36 -2x)x = x(4x² -144x +1296)

The volume will be a maximum where dV/dx = 0.

  dV/dx = 12x^2 -288x +1296 = 0

  x² -24x +108 = 0 . . . . divide by 12

  (x -6)(x -18) = 0 . . . . . factor

  x = 6 or 18 . . . . . . x = 18 gives a minimum volume; we want x = 6

Then the dimensions are 36 -2(6) = 24 inches square by 6 inches deep.

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