Final answer:
The minimum cost for constructing the box is determined by calculating the cost of the base, sides, and top using given prices per square foot for each, factoring in the fixed volume of 72 ft³. The dimensions that yield the minimum cost can be found by comparing two calculations for each possible option.
Step-by-step explanation:
To solve for the dimensions of the box that can be constructed at minimum cost, you need to consider the cost of materials for each part of the box (base, sides, top) and the constraint that the volume of the box must be 72 ft³. Since the box has a square base, if we let the side of the base be s in feet, the height h will be 72/s² ft due to the volume constraint. The total cost C is given by the sum of the costs for the base, sides (4 of them), and top:
C = (s² * $0.41) + (4 * s * h * $0.12) + (s² * $0.23)
Substitute h = 72/s² into the cost function and take the derivative to find the minimum cost. After findings that minimizes the cost, use that value to find h using the volume constraint. By calculating this for each option provided, we can determine which set of dimensions results in the lowest cost.
For option A (s = 6 ft), the cost would be:
- Base: 6 ft * 6 ft * $0.41/ft² = $14.76
- Sides: 4 sides * 6 ft * 2 ft * $0.12/ft² = $5.76
- Top: 6 ft * 6 ft * $0.23/ft² = $8.28
Total cost for option A = $14.76 + $5.76 + $8.28 = $28.80
Similar calculations would be done for options B, C, and D to find the minimum cost among them.