Question

an open-top box is to be made from a 36-inch by 42-inch piece of plastic by removing a square from each corner of the plastic and folding up the flaps on each side. what should be the length of the side x of the square cut out of each corner to get a box with the maximum volume? you may enter an exact answer or round to the nearest hundredth.

Answer 1

To find the length of the square cut out of each corner to get a box with the maximum volume, we can use calculus. By taking the derivative of the volume function and setting it equal to zero, we can solve for x to find the value that maximizes the volume.

Step-by-step explanation:

To find the length of the square cut out of each corner, we need to maximize the volume of the box. The volume of a rectangular box is equal to the length times the width times the height.

Let's assume that the length of each side of the square cut out of each corner is x. Then, the length of the box will be (36 - 2x) and the width will be (42 - 2x). The height will be x.

Therefore, the volume of the box can be expressed as:

V = (36 - 2x)(42 - 2x)x

To find the maximum volume, we can take the derivative of the volume function with respect to x, set it equal to zero, and solve for x. This will give us the value of x that maximizes the volume.

After finding the value of x, we can substitute it back into the volume function to find the maximum volume.

Answer 2

To find the maximum volume of the box, we can use calculus to determine the value of x that maximizes the volume function. By taking the derivative of the volume function and setting it equal to zero, we can solve for x. The only valid solution is x = 6 inches.

Step-by-step explanation:

To find the dimensions of the square to be cut out, we need to maximize the volume of the box. Let's assume the length of each side of the small square to be x inches.

When the flaps are folded up, the length of the box would be (42 - 2x) inches, and the width would be (36 - 2x) inches. The height of the box would be x inches. The volume of the box is given by V = length x width x height.

V = (42 - 2x)(36 - 2x)(x) = 4x³ - 156x² + 1512x.

To find the maximum volume, we need to find the value of x that maximizes the function V. We can do this by taking the derivative of V and setting it equal to zero: dV/dx = 12x² - 312x + 1512 = 0.

Solving this equation gives us x = 6 or x = 18. Since x must be less than half the length of the side of the plastic sheet (which is 36 inches), the only valid solution is x = 6 inches.

Therefore, the length of the side of the square to be cut out from each corner should be 6 inches.