Final answer:
To minimize the cost for plating a jewelry box with a square base and a given volume, create a cost function based on the surface area to be plated. Calculate the derivative, find the critical points, and use the second derivative test to find minimum cost dimensions. The side length 'x' of the base, and height 'h' can then be determined from the volume equation.
Step-by-step explanation:
To minimize the cost of materials for a jewelry box with a square base, a volume of 52 cm³, silver plated sides, and nickel plated top and bottom, we need to determine the dimensions that will result in the lowest surface area for plating because plating costs are based on area. Let's denote the side of the square base as 'x' cm, and the height of the box as 'h' cm. Given that the volume, V, is 52 cm³, we have V = x²h = 52, which means h = 52/x².
The total surface area, S, to be plated is the sum of the areas of the four sides (4xh) and the top and bottom (2x²). Substituting h, we get S = 4x(52/x²) + 2x². The cost function, C, will be C = 9(4xh) + 2(1x²), because the sides are silver plated at 9 per cm² and the top and bottom are nickel plated at 1 per cm².
To minimize the cost, take the derivative of the cost function with respect to 'x' and set it equal to zero to find the critical points. The minimum cost will occur at one of these points. Finally, use the second derivative test to confirm that it is indeed a minimum cost value. The critical point that yields the minimum cost will give you the dimensions of the box: the value of 'x' for the base and 'h' for the height can be calculated from the volume equation.