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

Question

2 years ago

14

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

1 answer:

seraphim [82] · Answer 1 · 2023-06-20T01:18:48+03:00

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

<h3>How to solve the dimension?</h3>

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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