5. (15 points) Optimization: I want to build a box to put my stuff where the total surface area is 16 square inches. To get max volume, what are the dimensions of the box?

Step 1 ：We are given a box with a total surface area of 16 square inches. We want to find the dimensions of the box that will maximize its volume.

Step 2 ：The box is a rectangular prism, so its volume is length * width * height, and its surface area is 2*(length*width + width*height + height*length).

Step 3 ：Assuming the box is a cube (all sides are equal), which is the most efficient shape for a box (maximizes volume for a given surface area), we can express width and height in terms of length (width = length, height = length).

Step 4 ：Substituting these into the surface area equation to get an equation in terms of length only: \(surface\_area = 6*length^2\)

Step 5 ：Solving this equation gives two possible values for length: \([-2*\sqrt{6}/3, 2*\sqrt{6}/3]\). The length of a box cannot be negative, so we discard the negative root.

Step 6 ：The positive root gives the length of the box. The volume is then calculated by cubing the length: \(volume = 16*\sqrt{6}/9\)

Step 7 ：\(\boxed{Final Answer: The dimensions of the box that will give the maximum volume under the condition of having a total surface area of 16 square inches are length = width = height = \frac{2\sqrt{6}}{3} inches. The maximum volume of the box is \frac{16\sqrt{6}}{9} cubic inches.}\)