Optimization

Step 1: Define the Variables
Let: - be the side length of the square base, - be the height of the box. The total surface area of the box is expressed as: Step 2: Solve for in Terms of
From the surface area equation: Step 3: Maximize the Volume
The volume of the box is given by: Differentiate the volume function with respect to : Set the derivative equal to zero to find the critical points: Step 4: Calculate
Substitute into the equation for : Final Answer: cm, cm.