Question

A metal box (without a top) is to be constructed from a square sheet of metal that is 20 cm on a side by cutting square pieces of the same size from the corners of the sheet and then folding up the sides. Find the dimensions of the box with the largest volume that can be constructed in this manner and prove that these values give a maximum volume.

Answer 1

To find the dimensions of the box with the largest volume, we need to consider the relationship between the length of the cut squares and the resulting dimensions of the box. By finding the critical points of the volume equation and determining which one gives the maximum volume, we can find the dimensions for the box with the largest volume and prove that these values correspond to a maximum.

Step-by-step explanation:

To find the dimensions of the box with the largest volume, we need to consider the relationship between the length of the cut squares and the resulting dimensions of the box. Let's assume that each side of the cut square has a length of 'x' cm. When folding up the sides, the resulting height of the box is 'x' cm. The resulting length of the box is (20-2x) cm, and the resulting width is (20-2x) cm as well.

To find the box's volume, we multiply the length, width, and height: V = (20-2x)(20-2x)x cm³

To find the dimensions for the box with the largest volume, we need to find the value of 'x' that maximizes this equation. We can do this by finding the critical points of the equation and determining which one gives the maximum volume. We can then prove that the values of 'x' that give this maximum volume indeed correspond to a maximum using calculus.