Find the volume of the largest box with no top that can be made from a piece of cardboard 30 inches square by cutting equal squares from the corners and turning up the sides. A) 2000 in³ B) 125 in³ C) 3000 in³ D) 1000 in³

Question

Quizplus · Accepted Answer

The volume of the box is V = x^3, where x is the length of the side of the square that is cut from each corner. To maximize the volume, we need to maximize x. The perimeter of the cardboard is 120 inches, so the perimeter of the square that is cut from each corner is 120 - 4x. The area of the square that is cut from each corner is x^2, so the area of the cardboard that is left is (30 - 2x)^2. The volume of the box is V = x^3 = (30 - 2x)^2 * x = 900x - 120x^2 + 4x^3. To find the maximum, we take the derivative of V with respect to x and set it equal to 0: dV/dx = 12x^2 - 240x + 900 = 0. Solving for x, we get x = 10. Therefore, the volume of the largest box is V = 10^3 = 1000 cubic inches.

Find the Volume of the Largest Box with No Top