MATH 131 Final Exam June 29,2023 6. A rectangular box with a square base, an open top, and a volume of $216 \mathrm{in}^{3}$ 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 that the volume of the box is \(V = x^2 * h\), where \(x\) is the side length of the square base and \(h\) is the height of the box.

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

Step 3 ：We are given that \(V = 216\), so we can solve for \(h\) in terms of \(x\) and substitute this into the surface area formula to get \(A = x^2 + 864/x\).

Step 4 ：Taking the derivative of the surface area with respect to \(x\), we get \(A' = 2*x - 864/x^2\).

Step 5 ：Setting this equal to zero and solving for \(x\), we get the critical points \(x = 6*2^{1/3}\), \(-3*2^{1/3} - 3*2^{1/3}*\sqrt{3}*I\), and \(-3*2^{1/3} + 3*2^{1/3}*\sqrt{3}*I\).

Step 6 ：Since the dimensions of a box cannot be complex, we discard the complex solutions. The real solution gives the side length of the square base.

Step 7 ：Substituting this value back into the formula for \(h\) gives the height of the box as \(h = 216/(6*2^{1/3})^2\).

Step 8 ：Substituting the dimensions into the surface area formula gives the minimum surface area as \(A = 108*2^{2/3}\).

Step 9 ：\(\boxed{\text{Final Answer: The dimensions of the box that minimize the surface area are a square base with side length approximately } 6*2^{1/3} \text{ inches and height approximately } 216/(6*2^{1/3})^2 \text{ inches. The minimum surface area is approximately } 108*2^{2/3} \text{ square inches.}}\)