A rectangular tank that is \( 6912 \mathrm{ft}^{3} \) with a square base and open top is to be constructed of sheet steel of a given thickness. Find the dimensions of the tank that minimizes the amount of material used. The dimensions of the tank with minimum weight are \( \square \mathrm{ft} \).

To minimize the amount of material used for a rectangular tank with a square base and an open top, we need to find the dimensions that minimize the surface area while maintaining a fixed volume. Let \( x \) be the length of one side of the square base, and \( h \) be the height of the tank. The volume \( V \) of the tank is given by: \[ V = x^2 h \] We know that the volume is \( 6912 \, \text{ft}^3 \), so we have: \[ x^2 h = 6912 \] From this, we can express \( h \) in terms of \( x \): \[ h = \frac{6912}{x^2} \] Next, we need to find the surface area \( S \) of the tank, which consists of the area of the base and the area of the four sides. The surface area is given by: \[ S = x^2 + 4xh \] Substituting \( h \) from the volume equation into the surface area equation: \[ S = x^2 + 4x\left(\frac{6912}{x^2}\right) \] Simplifying this, we get: \[ S = x^2 + \frac{27648}{x} \] To minimize the surface area, we take the derivative of \( S \) with respect to \( x \) and set it to zero: \[ \frac{dS}{dx} = 2x - \frac{27648}{x^2} \] Setting the derivative equal to zero: \[ 2x - \frac{27648}{x^2} = 0 \] Rearranging gives: \[ 2x = \frac{27648}{x^2} \] Multiplying both sides by \( x^2 \): \[ 2x^3 = 27648 \] Dividing by 2: \[ x^3 = 13824 \] Taking the cube root: \[ x = \sqrt[3]{13824} = 24 \, \text{ft} \] Now, substituting \( x \) back into the equation for \( h \): \[ h = \frac{6912}{x^2} = \frac{6912}{24^2} = \frac{6912}{576} = 12 \, \text{ft} \] Thus, the dimensions of the tank that minimize the amount of material used are: \[ \text{Base side length } x = 24 \, \text{ft}, \quad \text{Height } h = 12 \, \text{ft} \] The dimensions of the tank with minimum weight are \( 24 \, \text{ft} \) (base side length) and \( 12 \, \text{ft} \) (height).