Question

From a 50cm by 50cm sheet of aluminum, square corners are cut out so that the sides can be folded up to make a box. Determine the dimensions of the box to yield a box of maximum volume.

Answer 1

To determine the dimensions of the box that will yield the maximum volume, we can differentiate the volume formula and find the value of x that makes dV/dx equal to zero. The dimensions of the box that yield the maximum volume are (50-2(0.25))cm by (50-2(0.25))cm.

Step-by-step explanation:

To determine the dimensions of the box that will yield the maximum volume, we need to find the dimensions of the square corners that need to be cut out.

Let's assume that the length of each side of the square corners is 'x'.

The remaining unfolded portion will have dimensions (50-2x)cm by (50-2x)cm.

The volume of the box is given by V = x(50-2x)^2.

To find the dimensions of the box that yield the maximum volume, we can differentiate V with respect to x and set it equal to zero.

dV/dx = 0 => 100x - 200x^2 = 0.

Solving for x, we find that x = 0 or x = 0.25.

Since we can't have a box with no dimensions, we take x = 0.25.

Therefore, the dimensions of the box that yield the maximum volume are (50-2(0.25))cm by (50-2(0.25))cm.