Final answer:
The area of a larger square with side lengths twice as long as those of a smaller square is 4 times greater than the area of the smaller square because the ratio of the areas of similar figures is the square of the scale factor.
Step-by-step explanation:
The question involves understanding the properties of similar shapes and the relationship between their dimensions and areas. To find how the area of a larger square compares to a smaller square when the side lengths are related by a scale factor, we can use the rule that the ratio of the areas of similar figures is the square of the scale factor. Given that the larger square has dimensions that are twice as large as those of the smaller square, the scale factor is 2.
First, calculate the side length of the larger square:
Side length of larger square = 4 inches × 2 = 8 inches.
Then, compare the two areas:
Area of the smaller square = 4 inches × 4 inches = 16 square inches.
Area of the larger square = 8 inches × 8 inches = 64 square inches.
The ratio comparing the two areas is 64:16, which simplifies to 4:1. Therefore, the area of the larger square is 4 times greater than the area of the smaller square.