A rectangular tank with a square base, an open top, and a volume of $2916 \mathrm{ft}^{3}$ is to be constructed of sheet steel. Find the dimensions of the tank that has the minimum surface area
The dimensions of the tank with minimum surface area are $\mathrm{ft}$. (Simplify your answer Use a comma to separate answers )

Step 1 ：The problem is asking for the dimensions of a rectangular tank with a square base that has the minimum surface area. The volume of the tank is given as 2916 cubic feet.

Step 2 ：The volume of a rectangular tank with a square base is given by the formula \(V = x^2 * h\), where \(x\) is the length of the side of the square base and \(h\) is the height of the tank.

Step 3 ：The surface area of the tank (excluding the open top) is given by the formula \(A = x^2 + 4*x*h\).

Step 4 ：We want to minimize the surface area \(A\) while keeping the volume \(V\) constant at 2916 cubic feet.

Step 5 ：This is a problem of optimization with constraints. We can solve it using calculus.

Step 6 ：First, we can express \(h\) in terms of \(x\) using the volume equation: \(h = V / x^2 = 2916 / x^2\).

Step 7 ：Then we can substitute \(h\) into the surface area equation to get \(A\) in terms of \(x\) only: \(A = x^2 + 4*x*(2916 / x^2) = x^2 + 11664 / x\).

Step 8 ：We can find the minimum of \(A\) by taking the derivative of \(A\) with respect to \(x\), setting it equal to zero, and solving for \(x\).

Step 9 ：The critical points are 18, -9 - 9*sqrt(3)*I, and -9 + 9*sqrt(3)*I. The negative and complex solutions are not meaningful in this context, so we discard them. The only meaningful solution is \(x = 18\).

Step 10 ：We can substitute \(x = 18\) into the equation for \(h\) to find the height of the tank.

Step 11 ：The height of the tank is 9 feet.

Step 12 ：So, the dimensions of the tank that minimize the surface area are a base of 18 feet by 18 feet and a height of 9 feet.

Step 13 ：Final Answer: The dimensions of the tank with minimum surface area are \(\boxed{18, 18, 9}\) feet.