A rectangular box with a square base, an open top, and a volume of 216 in^3 is to be constructed. What should the dimensions of the box be to minimize the surface area of the box? What is the minimum surface area?.

Step 1 ：Given a rectangular box with a square base, an open top, and a volume of 216 in^3. We need to find the dimensions of the box that minimize the surface area.

Step 2 ：The volume of the box 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 box.

Step 3 ：The surface area of the box is given by the formula \(A = x^2 + 4*x*h\). We want to minimize \(A\) while keeping \(V\) constant at 216.

Step 4 ：We can express \(h\) in terms of \(x\) and \(V\) using the volume formula: \(h = V/x^2 = 216/x^2\).

Step 5 ：Substitute \(h\) into the surface area formula to get \(A = x^2 + 864/x\).

Step 6 ：Take the derivative of \(A\) with respect to \(x\), set it equal to zero, and solve for \(x\). The derivative is \(2*x - 864/x^2\).

Step 7 ：Solving the equation gives three solutions for \(x\): \(6*2^{1/3}\), \(-3*2^{1/3} - 3*2^{1/3}*\sqrt{3}*I\), and \(-3*2^{1/3} + 3*2^{1/3}*\sqrt{3}*I\). Only the first solution is a real positive number, so it is the only valid solution.

Step 8 ：Substitute \(x = 6*2^{1/3}\) into the surface area formula to find the minimum surface area: \(A = 108*2^{2/3}\).

Step 9 ：Substitute \(x = 6*2^{1/3}\) into the height formula to find the corresponding height: \(h = 3*2^{1/3}\).

Step 10 ：\(\boxed{\text{Final Answer: The dimensions of the box that minimize the surface area are a square base with side length approximately } 6*2^{1/3} \text{ inches and height approximately } 3*2^{1/3} \text{ inches. The minimum surface area is approximately } 108*2^{2/3} \text{ square inches.}}\)