Final answer:
The dimensions of the box that minimize the cost are 2 feet by 2 feet by 3 feet. We derive this by creating a cost function based on the volume constraint and the material costs, and then use calculus to find the minimum.
Step-by-step explanation:
To solve the mathematical problem of constructing a closed rectangular box with a square base and a volume of 12 cubic feet with the least cost, we need to apply calculus to minimize the total cost of the materials used. Let's let the side of the square base be x feet, and the height be h feet. The volume of the box is then x2h = 12.
We can express h as h = 12/x2. When we calculate the surface area, the base does not contribute to the cost since it only mentions the sides and the top. The cost of the wood sides will be 4xh, and the cost of the metal top is x2. Thus, the total cost function C is C(x) = 4xh + 2x2 = 4x(12/x2) + 2x2 = 48/x + 2x2.
To find the minimum cost, we take the derivative of the cost function with respect to x and set it to zero. The derivative, dC/dx = -48/x2 + 4x. Setting dC/dx to zero we get 4x3 = 48, which implies that x3 = 12, thus x = 2. The third derivative test confirms a minimum. Since h is dependent on x, we have h = 12/x2 = 3 feet. Therefore, the dimensions of the box that minimize the cost are 2 feet by 2 feet by 3 feet.