Find the volume of the largest box with no top that can be made from a piece of cardboard 60 inches square by cutting equal squares from the corners and turning up the sides. A) 16,000 in³ B) 1000 in³ C) 24,000 in³ D) 8000 in³

Question

Quizplus · Accepted Answer

The volume of the box is given by 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 60 inches, so the perimeter of the square that is cut from each corner is 60 - 4x inches. The area of the square that is cut from each corner is (60 - 4x)^2 square inches. The area of the cardboard that is left after the squares are cut out is 60^2 - 4(60 - 4x)^2 square inches. The volume of the box is V = x^3 = (60^2 - 4(60 - 4x)^2)^3/27 cubic inches. To maximize the volume, we need to find the value of x that maximizes the expression (60^2 - 4(60 - 4x)^2)^3/27. We can do this by taking the derivative of the expression with respect to x and setting it equal to 0. The derivative is dV/dx = 4x(60^2 - 4(60 - 4x)^2)^2/27. Setting this equal to 0, we get 4x(60^2 - 4(60 - 4x)^2)^2/27 = 0. This implies that either x = 0 or (60^2 - 4(60 - 4x)^2)^2 = 0. If x = 0, then the volume of the box is 0. If (60^2 - 4(60 - 4x)^2)^2 = 0, then 60^2 - 4(60 - 4x)^2 = 0. Solving for x, we get x = 15. Therefore, the maximum volume of the box is V = (60^2 - 4(60 - 4(15))^2)^3/27 = 16,000 cubic inches.

Find the Volume of the Largest Box with No Top