Question

A jewelry box with a square base is to be built with silver plated sides, nickel plated bottom and top, and a volume of 28 cm³. If nickel plating costs 1 per cm² and silver plating costs8 per cm², find the dimensions of the box to minimize the cost of the materials.

Answer 1

To minimize the cost of materials for the jewelry box, we need to minimize the total surface area. We can find the dimensions that minimize the cost by taking the derivative of the cost function and solving for x. The dimensions that minimize the cost are 6 cm by 6 cm by 1.55 cm.

Step-by-step explanation:

To minimize the cost of materials for the jewelry box, we need to minimize the total surface area, since that is where the cost is incurred. Let's assume the dimensions of the base of the box are x by x. The height of the box is then 28/x^2. The surface area of the sides is 4x * 28/x^2 = 112/x and the surface area of the top and bottom is 2x^2.

Therefore, the total cost of silver plating is 8 * (112/x) and the total cost of nickel plating is 1 * 2x^2. The total cost function is C(x) = 8 * (112/x) + 2x^2.

To minimize the cost, we take the derivative of the cost function with respect to x, set it equal to zero, and solve for x:

C'(x) = -896/x^2 + 4x = 0

Multiplying through by x^2, we get -896 + 4x^3 = 0

Dividing by 4, we get x^3 = 224

Taking the cube root of both sides, we get x = 6

So the dimensions of the box that minimize the cost of materials are 6 cm by 6 cm by 28/(6^2) cm.