Let x be the length of each side of the square cut from each corner.
The dimensions of the box after cutting the squares would be:
Length = 14 - 2x
Width = 8 - 2x
Height = x
The volume of the box V can be calculated as:
V = (14 - 2x)(8 - 2x)x
V = 112x - 36x^2 + 4x^3
To find the maximum volume, we need to find the derivative of V with respect to x and set it equal to zero:
dV/dx = 112 - 72x + 12x^2
0 = 12x^2 - 72x + 112
0 = x^2 - 6x + 14/3
Using the quadratic formula, we find:
x = (6 ± √(6^2 - 4(1)(14/3))) / 2
x = (6 ± √(36 - 56/3)) / 2
x = (6 ± √(108/3 - 56/3)) / 2
x = (6 ± √(52/3)) / 2
x = (6 ± 2√13/3) / 2
Since x must be positive, we take the positive value:
x = (6 + 2√13/3) / 2
x = 3 + √13/3
Therefore, the size of the square cut from each corner for the box to have the maximum volume is approximately 3 + √13/3 inches.
A rectangular piece of cardboard, 8 inches by 14 inches, is used to make an open top box by
cutting out a small square from each corner and bending up the sides. What size square should
be cut from each corner for the box to have the maximum volume?
1 answer