Final Answer:
The dimensions of the box that minimize the total cost are a square base of side length a = ∛108 meters and a height h = 108 / a² meters.
Step-by-step explanation:
To minimize the cost, we need to set up an expression for the total cost in terms of the dimensions of the box. The cost consists of the bottom and sides. The bottom is a square, so its cost is 40a², and the sides have an area of 4ah, resulting in a cost of 30 * 4ah. The total cost function is C(a, h) = 40a² + 120ah. Since the volume of the box is fixed at 108m³, we have the constraint V = a²h = 108.
Solving for h, we get h = 108 / a². Substituting this into the cost function, we have C(a) = 40a² + 4320 / a. To find the minimum, take the derivative of C with respect to a, set it equal to zero, and solve for a. The critical point is a = ∛108. Substituting this back into the equation for h, we find the corresponding h.
In conclusion, the dimensions that minimize the total cost are found by minimizing the cost function subject to the constraint of a fixed volume. The critical point is determined by setting the derivative of the cost function equal to zero, leading to the dimensions a = ∛108 and h = 108 / a².