Final answer:
The student question involves finding the dimensions of a box with a square base that has a fixed volume and the smallest possible surface area, using the formulas for volume and surface area of a cuboid with a square base.
Step-by-step explanation:
The question pertains to the optimization of a box with a square base of side x and height h, given the constraints of a fixed volume and minimal surface area. The total volume of the box V is given as V = x²h, which is the formula for the volume of a cuboid (in this case, a cube since the base is square) where the area of the base (A) is x², and h is the height. The surface area S of a box with a square base is given by S = 2x² + 4xh (two square base areas plus the areas of the four sides).
Since the volume is constant at 15, we have the equation x²h = 15. To minimize the surface area, we need to find the dimensions that lead to the smallest possible surface area. Taking the derivative of the surface area equation with respect to x, and setting it equal to zero while considering h as a function of x due to the volume constraint, will provide the optimal dimensions that satisfy the requirements. Through this process, the dimensions that yield a volume of 15 with the smallest surface area can be determined.