Final answer:
Mrs. Carter can make three distinct rectangles with her 18 boxes of tacks, with dimensions of 1x18, 2x9, or 3x6. For Marta's squares, the larger one with side length of 8 inches has an area four times greater than the smaller 4-inch square.
Step-by-step explanation:
To determine the different rectangles Mrs. Carter can make with 18 boxes of tacks, we need to consider the factors of 18. The factors of 18 are pairs of numbers that multiply together to give 18. These factors give us the possible dimensions for the rectangles, which are combinations of length (l) and width (w) where l × w = 18.
The possible dimensions (l × w) for the rectangles are:
- 1 × 18 (a long, narrow rectangle)
- 2 × 9 (a more balanced rectangle)
- 3 × 6 (a rectangle that is closer to a square shape)
- 6 × 3 (this has the same dimensions as 3 × 6, so it's not a distinct rectangle)
- 9 × 2 (this has the same dimensions as 2 × 9, so it's not a distinct rectangle)
- 18 × 1 (this has the same dimensions as 1 × 18, so it's not a distinct rectangle)
Considering these pairs, we can see there are three distinct rectangles that can be made with these dimensions.
Comparing the Area of Two Squares
For the problem involving the squares, if Marta has a square with a side length of 4 inches and a similar square that is twice the dimensions of the first one, the side length of the larger square will be 8 inches. Since the area of a square is found by squaring the side length, the area of the smaller square is 16 square inches (4 inches × 4 inches), and the area of the larger square is 64 square inches (8 inches × 8 inches).
The area of the larger square is four times the area of the smaller square because the scale factor here applies to the length and width, thus it is squared when calculating the area.