Final answer:
The dimensions of Carmen's square picture without the border are 8 by 8 inches when x equals 0 in the area function A(x) = (8 + 2x)^2.
Step-by-step explanation:
To find the dimensions of Carmen's picture without the border, we look at the function A(x) = (8 + 2x)^2. This represents the area of the picture with a border x inches wide. Without a border (x=0), the dimensions of the picture are simply the square root of the area when x=0. Thus, the area without the border is 82 or 64 square inches. This means the picture without the border is 8 by 8 inches.
The answer is C. 8 by 8 inches. When comparing the side lengths of two similar squares, like in Marta's squares, the scale factor between the sides squared will give the ratio of their areas. With a side length of 4 inches for the smaller square and side lengths double for the larger square, the larger square has sides that are 8 inches, thus the area is four times larger.
Understanding this concept of scale factor and area is key in solving problems involving similar figures and their dimensions or areas, whether it's with squares, rectangles, or other shapes.