Question

An open-reinforced box is made from a 12- by 36 -inch piece of cardboard by cutting along the marked (green) lines, discarding the shaded pieces, and folding as shown in the figure (below). What size squares should be cut from the corners in order to have a box with maximum volume? dustify your answer by displaying all work.

Answer 1

To find the size of squares that should be cut from the corners to have a box with maximum volume, follow these steps:

Step-by-step explanation:

To find the size of squares that should be cut from the corners to have a box with maximum volume, we need to follow these steps:

1. First, we need to determine the dimensions of the box when it is constructed. The length will be the original length of the cardboard minus two times the size of the squares cut from the corners. Similarly, the width will be the original width of the cardboard minus two times the size of the squares. The height will be the size of the squares cut from the corners.
2. Next, we need to express the volume of the box in terms of the size of the squares. The volume of the box is given by the formula:
   `V = length * width * height.`
3. Then, we can substitute the expressions for the length, width, and height into the volume formula and simplify the equation.
4. Finally, we can differentiate the equation with respect to the size of the squares, set the derivative equal to zero, and solve for the size of the squares that maximizes the volume. This will give us the size of the squares that should be cut from the corners to have a box with maximum volume.