Final answer:
The student wants to find the minimum cost to construct a rectangular box with a volume of 170 cm³. To minimize cost, the best approach is to assume the shape of the box is a cube, calculate its surface area, and then apply the given costs for top/bottom and sides.
Step-by-step explanation:
The student is asking to find the minimum cost of constructing a rectangular box with a given volume, where the top and bottom and sides have different costs per unit area. We are provided with the volume of the box (170 cm3), and the cost per cm2 of the top/bottom (3 cents) and the sides (9 cents). To minimize cost, we need to find the dimensions of the box that give us the smallest surface area, because cost is dependent on the surface area.
The box will have a length l, width w, and height h, and the surface area S can be expressed as S = 2lw + 2lh + 2wh. Since we want to minimize S to minimize cost, we need to consider that lwh = 170 cm3 (the volume of the box), which is a constraint on our dimensions. Using calculus, we can set up a function for cost in terms of l, w, and h, and use l and w in terms of h or vice versa to find the minimum.
However, the student did not ask for the specific dimensions, only the cost, so we alternatively can reason that the box with the minimum surface area for a given volume will be a cube. A cube has equal sides, so we can find the side length s of the cube by taking the cube root of the volume: s = ∓(170 cm3). We then calculate the surface area of the cube and multiply by the respective costs for the top/bottom and sides.