To determine the volume of the box, we need to find the length, width, and height of the box. We can do this by subtracting twice the length of the square cutouts from the original length, twice the width of the square cutouts from the original width, and the height will be the length of the square cutouts.
Let's assume that the side length of the square cutouts is x. Then the length of the box is (24 - 2x), the width of the box is (20 - 2x), and the height of the box is x. Thus, the volume of the box can be expressed as:
V(x) = (24 - 2x) * (20 - 2x) * x
To simplify this expression, we can expand it using the distributive property and then combine like terms:
V(x) = 4x^3 - 88x^2 + 480x
The approximate domain for V(x) would be [0, 10] since the side length of the square cutouts cannot be greater than half the length or width of the original piece of metal, which is 12 and 10 respectively.
To calculate V(x), we simply substitute x into the formula for V(x):
V(x) = 4x^3 - 88x^2 + 480x
Let's say we want to calculate V(2):
V(2) = 4(2)^3 - 88(2)^2 + 480(2)
= 16 - 352 + 960
= 624
Therefore, the volume of the box when the side length of the square cutouts is 2 inches is 624 cubic inches.