Final answer:
To minimize the cost of constructing a jewelry box with a square base, copper-plated sides, nickel-plated top and bottom, and a volume of 40 cm³, one must solve an optimization problem using calculus to find the dimensions that minimize the total plating cost.
Step-by-step explanation:
The student is trying to determine the dimensions of a jewelry box that will minimize the cost of the materials given that the box has a square base, a fixed volume of 40 cm³, and is made with different materials for different parts of the box. The cost of the materials varies depending on the type of plating; copper plating costs 1 per cm³ and nickel plating costs 2 per cm³.
To minimize the cost, we need to minimize the surface area because cost is directly dependent on the surface area that is plated. Let's denote the side of the square base as 's' and the height of the box as 'h'. Since the volume of the box, V, is 40 cm³, we have the equation s²h = 40. We need to express the cost in terms of a single variable by substituting h = 40/s² into the cost equation. The total cost of plating the box will be the cost of plating the four sides with copper and the top and bottom with nickel, given by C = copper cost + nickel cost = 4(s²) + 2(s²), where 4 is the number of sides and 2 represents the top and bottom areas.
By substituting h with 40/s² in the cost equation, we can take the derivative with respect to s, set it equal to zero, and solve for s to find the minimum cost. This problem is an example of an optimization problem in calculus where we determine the dimensions that will yield the minimum cost of construction.