Final answer:
Marcus's assumption about the surface area of prisms is incorrect; it does not account for the actual shape differences between prisms. Doubling the dimensions of a square quadruples its area. The relationship between dimensions and area is not as straightforward as presumed when comparing different shapes.
Step-by-step explanation:
I disagree with Marcus's assumption that the surface area of a right triangular prism is half that of a right rectangular prism, and using the formula SA=1/2(2lw+2lh+2wh) would be incorrect for calculating the surface area of a triangular prism. Surface area calculations must consider the actual shapes that make up the faces of the prism, and a triangular prism has different shaped faces compared to a rectangular prism.
Let's take Marta's squares as an example to illustrate how changing dimensions affects area. Marta has one square with side length of 4 inches. A second square has sides twice as long, which means the side length is 8 inches. The area of a square is calculated by squaring the side length, so the smaller square has an area of 4 inches × 4 inches = 16 square inches. For the larger square, the area is 8 inches × 8 inches = 64 square inches. Thus, when the dimensions are doubled, the area increases by a factor of four (because 2² is 4), not two.
The confusion might arise from considering linear dimensions versus area or volume. While doubling the dimensions of a 3D object like a cube will indeed quadruple the cross-sectional area (as it's essentially a 2D measurement), the volume will increase by a factor of eight, because 2² × 2 is 8. Therefore, Marcus's assumption does not hold when considering different dimensions and shapes.