Answer:
192.5 in³
Step-by-step explanation:
The cardboard is 10 by 20 before removing a square from each end. Assuming that the square is x inches wide. Therefore, the 20 in side gets reduced by x inches on both sides, or say it becomes 20 - 2x inches. On the other hand, the 10 in side is also reduced by 2x. The x value we get happens to be the height of the box when the sides are folded up.
Thus the volume V = lbh =
V = (20-2x)*(10-2x)*(x)
V = 4x³ - 60x² + 200x
On differentiating, we have dv/dx to be
dv/dx = 12x² - 120x + 200
Using general formula to find the roots of this equation, we can solve that x = 7.886 and x = 2.113
This roots we got are possible values of x, the square we cut. Since 7.886 * 2 = 15.772 inches, this is more than the 10 inch side, henceforth x = 2.113 inches.
You cut 2.113 inches from each corner to obtain the maximum volume.
The sizes of the cubes are
20 - (2 * 2.113) = 15.774
10 - (2 * 2.113) = 5.774
2.113
The volume of the cube is 15.774 * 5.774 * 2.113 = 192.5 cubic inches.
