Find the demension of the box of maximum value.

Question

An open box is made from a square piece material with sides 2 feet by cutting equal square from each corner. Find the demension of the box of maximum volume.

Answer 1

Let

:

x = the dimension of the square to be cut.

(2 -2x) = the side of the box

((2-2x)^2)x = V

(4 - 8x + 4x^2)x = V

4x -8x^2 + 4x^3 = V

dV/dx = 4 - 16x + 12x^2 = 0

1 -4x + 3x^2 = 0

x^2 - (4/3)x + (2/3)^2 = -1/3 +(2/3)^2

(x - 2/3)^2 = -3/9 + 4/9 = 1/9

x =2/3 +or - 1/3 = 1, 1/3

If x = 1 , (2 -2x) =0 hence no box can be formed if x = 1

If x = 1/3, (2- 2/3) = 4/3

The box therefore has sides of 4/3 feet and a height of 1/3 foot

Answer 2

old problem, although usually it's a rectangular piece.

let x = one side of the cutout, and also the height of the box

After folded, the side of the box is 2-2x

volume = (2-2x) (2-2x) x

V = 4x (x-1)² = 4x (x² -2x +1) = 4x³ -8x² +4x

dV/dx = 12x² - 16x + 4 = 0

3x² - 4x + 1 = 0

(3x - 1)(x - 1) = 0

x = 1/3 or 1

1 doesn't work

x = 0.333 ft

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