Final answer:
The largest possible volume of the box is obtained through an optimization problem where the surface area (SA = x² + 4xh = 10,800 cm²) is a constraint, and the volume (V = x²h) is the function to be maximized by expressing 'h' in terms of 'x', differentiating the volume function, and finding its maximum.
Step-by-step explanation:
To find the largest possible volume of a box with a square base and an open top when 10,800 cm² of material is available, we can form an optimization problem. Let the sides of the square base be x cm and the height h cm. We then have the surface area of the four sides and the base as SA = x² + 4xh = 10,800 cm². The volume of the box is V = x²h. To find the maximum volume, we need to express h in terms of x using the surface area constraint and then find the volume function V(x). To optimize the volume, we would take the derivative of the volume function V'(x) and find its critical points by setting the derivative to zero, and verifying which point corresponds to a maximum by using the second derivative test or analyzing the sign changes of the first derivative.