Final answer:
To minimize the weight of a square-based tank with a volume of 256 ft³, calculate the height as the volume divided by the base area. The surface area, dependent on the base and the four walls, can be minimized to determine the optimal base dimension, and consequently, the minimal weight of the steel for construction.
Step-by-step explanation:
To find the dimensions of a rectangular tank with a square base and an open top that result in the minimum weight, we should first establish that the volume of this tank is given as 256 ft3. If the base has sides of length x feet, then the area of the base is x2 ft2. The height h of the tank can be determined by the volume, so h = 256/x2 ft.
Now, the surface area of the tank is the area that will be covered by the sheet steel, and the sheet steel's weight will be proportional to this surface area. The open top means we only need to consider the base and the four walls for surface area. The total surface area A for the three sides and the base is given by A = x2 + 4xh. Substituting h yields A = x2 + 4x(256/x2) = x2 + 1024/x.
To minimize weight, we need to minimize the surface area, which means we need to find the minimum value of A. Taking the derivative of A with respect to x and setting it to zero will give us the critical value for x that will minimize A. Solving this optimization problem using calculus, the dimensions that minimize the weight of the tank can be determined.