To minimize the total cost of material for the rectangular container, we need to find the dimensions that will minimize the surface area while satisfying the given volume requirement.
Let's assume the length of the square base is 'x' feet, and the height of the container is 'h' feet.
The volume of a rectangular container with a square base can be calculated as follows:
Volume = length * width * height
Since the base is square, the length and width are the same, so we have:
Volume = x * x * h = x^2 * h
Given that the volume is 2673 ft^3, we can write the equation:
x^2 * h = 2673
To minimize the total cost of material, we need to minimize the surface area of the container, which is composed of the top, bottom, and four sides.
The surface area can be calculated as:
Surface Area = 2(x * x) + 4(x * h)
Now we can express the surface area in terms of a single variable, x:
Surface Area = 2x^2 + 4xh
To minimize the total cost, we need to minimize the surface area. However, the given problem does not provide the cost function explicitly. Therefore, we are unable to directly optimize the total cost with respect to the surface area.
To solve this problem further and find the dimensions that minimize the total cost, we need additional information about the cost function, such as the relationship between the cost per square foot and the surface area or the specific cost associated with each component (top, bottom, sides).
Please provide more details about the cost function or any additional information for us to proceed with finding the dimensions that minimize the total cost of material.