The size of the squares should be 3 inches in order to create a box with the largest possible volume.
If squares of equal sizes are to be cut out from the four corners so the four sides can be bent upwards, then the length of these squares will be the height of the box. Also twice of this length subtracted from the initial length and width of the material will be the new length and width of the box, respectively.
let x = length of square to be cut = height of the box
length of the box = 30 - 2x
width of the box = 14 - 2x
The volume of the box, which is a rectangular prism, is the product of the length, height and width.
V = l x w x h
V = (30 - 2x)(14 - 2x)(x)
V = 420x - 88x² + 4x³
To obtain the largest volume possible, the first derivative of the volume should be equal to zero.
V'(x) = 0
V = 420x - 88x² + 4x³
420 - 176x + 12x² = 0
Simplify and solve for the value of x.
12x² - 176x + 420 = 0
3x² - 44x + 105 = 0
(3x - 35)(x - 3) = 0
x = 35/3 ; x = 3
Check each value.
When x = 35/3 = h
l = 30 - 2x = 20/3
w = 14 - 2x = -28/3 ( length should be positive)
When x = 3 = h
l = 30 - 2x = 24
w = 14 - 2x = 8
Hence, the length of square to be cut is 3 in.
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