Question

a box is constructed out of two different types of metal. the metal for the top and bottom, which are both square, costs $4 per square foot and the metal for the sides costs $7 per square foot. find the dimensions that minimize cost if the box has a volume of 30 cubic feet?

Answer 1

To minimize the cost of the box with a volume of 30 cubic feet, we need to minimize the total surface area. This can be done by optimizing a cost function based on the dimensions of the box.

Step-by-step explanation:

To minimize the cost of the box, we need to minimize the total surface area. Let's assume that the length, width, and height of the box are represented by x, y, and z, respectively. The volume of the box is given as 30 cubic feet, which means that xyz = 30.

The surface area of the box can be calculated by adding the areas of the top, bottom, and four sides. The cost of the metal for the top and bottom is $4 per square foot, so the cost for the top and bottom is 2 * 4 * (xy). The cost for the sides is $7 per square foot, so the cost for the sides is 4 * 7 * (xz + yz). Therefore, the total cost of the box can be represented as:

C = 2 * 4 * xy + 4 * 7 * (xz + yz)

Now, we need to optimize this cost function to find the dimensions that minimize the cost. To do this, we can take the derivative of C with respect to x, y, and z, set them equal to zero, and solve for x, y, and z. However, since this is a multi-variable optimization problem, it can be quite complex to solve analytically. Alternatively, we can use numerical optimization methods or software to find the values of x, y, and z that minimize the cost function.