Final answer:
To minimize the cost for a square box with a volume of 100ft³, one must find the side length of the box by calculating the cube root of the volume. With the side length determined as 4.64ft, the minimal total cost will be approximately $300.30 when using the given prices for base and side materials.
Step-by-step explanation:
To determine the dimensions that minimize the cost of a square box with a volume of 100ft³, we must consider the cost of materials for the base and sides separately. Given that the volume of a cube is the cube of its side length (s), we have s³ = 100ft³, leading to the side length s = ∛(100ft³) = 4.64ft. Now, the base area is s², and the total area of the four sides is 4s².
The base material costs $6 per square foot; thus, the base cost is 6s² dollars. The side material costs $2 per square foot; thus, the side cost is 4 × 2s² = 8s² dollars. To minimize the total cost, we add the costs to obtain the cost function: Total Cost = 6s² + 8s² = 14s². Given that s = 4.64ft, we find that the cost is minimized when the side length of the square base is 4.64ft. The Total Cost is 14 × (4.64)^2 = $300.30 (rounded to the nearest cent).
The minimal cost for creating a square box with a volume of 100ft³ is when each side of the base is 4.64ft, which equals a cost of approximately $300.30 for the base and sides together.