Final answer:
The surface area of a rectangular prism increases by a factor of the square of the scaling factor. Thus, increasing the prism's dimensions by a factor of 5 leads to an increase in the surface area by a factor of 25.
Step-by-step explanation:
The question involves understanding how the surface area of a geometric shape changes when its dimensions are scaled up. Initially, we have a rectangular prism with a length of 10.4 inches, width of 2 inches, and a height of 8.25 inches. To find the surface area of this prism, we would calculate the sum of the areas of all six faces.
However, the question focuses on what happens to the surface area when the dimensions of the prism are increased by a factor of 5. It might be easy to believe that the surface area will also increase by a factor of 5², or 25 times, since the dimensions are multiplied by 5. But, let's examine this more closely using mathematical reasoning. For a rectangular prism, the surface area (SA) is calculated using the formula: SA = 2lw + 2lh + 2wh, where l is the length, w is the width, and h is the height. When each dimension is multiplied by 5, the new surface area becomes SA = 2 × 5l × 5w + 2 × 5l × 5h + 2 × 5w × 5h, or SA = 25 × 2(lw) + 25 × 2(lh) + 25 × 2(wh), which is 25 times the original surface area. Therefore, the claim is true: increasing the dimensions of a rectangular prism by a factor of 5 will indeed increase the surface area by a factor of 25.