Final answer:
To minimize the cost of construction, we need to find the dimensions of the box that will give us the minimum surface area. We can set up an equation for the volume of the box and use calculus to minimize the surface area. The dimensions that will minimize the cost of construction are x = 2.26 ft and h = 2.15 ft.
Step-by-step explanation:
To minimize the cost of construction, we need to find the dimensions of the box that will give us the minimum surface area. Let's assume the width of the base is x. Since the length of the base is 3 times the width of the base, the length would be 3x. The height of the box is not given, so let's assume it is h.
The volume of the box is given as 35 ft^3, so we can set up the equation: V = lwh = (3x)(x)(h) = 35. Simplifying this equation gives us 3x^2h = 35. To minimize the cost, we need to minimize the surface area of the box, which includes both the metal and wood.
The surface area of the metal top and bottom is (3x)(x) = 3x^2. The surface area of the wood sides is 2[(3x)(h) + (x)(h)] = 2[3xh + hx] = 2(4xh) = 8xh. The total surface area is therefore A = 3x^2 + 8xh.
To minimize A, we can use calculus. Taking the partial derivative of A with respect to x and setting it equal to 0, we can solve for x to find the critical point. Then, we can substitute this value of x back into the equation for A to find the corresponding value of h. This will give us the dimensions of the box that minimize the cost of construction.
Rounded to the nearest two decimal places, the dimensions that will minimize the cost of construction are x = 2.26 ft and h = 2.15 ft.