Question
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?
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?
Answers
GPT 3.5
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.
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.