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An open box is to be made from a square piece of cardboard, 18 inches by 18 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

Respuesta :

The dimension of the box of the greatest volume that can be constructed in this way is 12x12x3 and the volume is 432.

How to solve the dimension?

Let x be the side of the square to remove. Then the volume of the box is:

V(x) = (18 - 2x)² * x = 324x - 72x² + 4x³

To find the maximum volume, differentiate and set it to 0:

V'(x) = 324 - 144x + 12x²

0 = x² - 12x + 27

0 = (x - 9)(x - 3)

x = 3 or 9

When x = 3,

V"(x) =-144+24x

V"(3) =-144+72=-72<0

so volume is maximum at x=3

Therefore the box is 12x12x3 and the volume is 432.

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