Answer:
The amount to remove from each corner is a square, 4/3 ft by 4/3 ft.
Step-by-step explanation:
The piece of cardboard measures 8 ft by 8 ft.
Each corner will be cut to create the sides of the box.
Let the square cut from each corner measure x by x ft.
The base of the box will measure 8 - 2x by 8 - 2x, and the height of the box will be x.
V = (8 - 2x)^2 * x
V = 64x - 32x^2 + 4x^3
Now we differentiate and set the derivative equal to zero to find a maximum value.
64 - 64x + 12x^2 = 0
3x^2 - 16x + 16 = 0
(3x - 4)(x - 4) = 0
3x - 4 = 0 or x - 4 = 0
3x = 4 or x = 4
x = 4/3 or x = 4
The solution x = 4 must be discarded since 8 - 2x = 8 - 8 = 0, and the sides would have 0 length.
The amount to remove from each corner is a square, 4/3 ft by 4/3 ft.
