A rectangular tank with a square base, an open top, and a volume of $19,652 \mathrm{ft}^{3}$ is to be constructed of sheet steel. Find the dimensions of the tank that has the minimum surface area.

Step 1 ：The problem is asking for the dimensions of a rectangular tank with a square base and an open top that has the minimum surface area. The volume of the tank is given as 19652 cubic feet.

Step 2 ：The volume V of a rectangular tank with a square base of side x and height h is given by \(V = x^2 * h\). We know that \(V = 19652\), so we can express h in terms of x as \(h = 19652 / x^2\).

Step 3 ：The surface area A of the tank (excluding the open top) is given by \(A = x^2 + 4*x*h\). Substituting \(h = 19652 / x^2\) into this equation gives \(A = x^2 + 4*x*(19652 / x^2) = x^2 + 78508 / x\).

Step 4 ：We want to minimize A. To do this, we can take the derivative of A with respect to x, set it equal to zero, and solve for x. This will give us the value of x that minimizes A. We can then substitute this value of x back into the equation for h to find the corresponding value of h.

Step 5 ：The critical points are complex numbers and a real number. The real number is the only possible solution since the dimensions of the tank cannot be complex. The real number is the cube root of 39254. The height is calculated by dividing the volume by the square of the dimension of the base.

Step 6 ：Final Answer: The dimensions of the tank that has the minimum surface area are \(\boxed{x = 39254^{1/3}}\) feet for the base and \(\boxed{h = \frac{9826*39254^{1/3}}{19627}}\) feet for the height.