Question

7 a) how did the scale factor of 4 change the area of the rectangle?
original area\tnew area\thow did it change?
4.5?
units?\t18??\t
7 b) how did the scale factor of 4 change the side lengths?

Explanation:

Step1: Recall Scale Factor for Area

For a rectangle, area \( A = l \times w \). If scale factor \( k = 4 \) is applied to length and width, new length \( l' = 4l \), new width \( w' = 4w \). New area \( A' = l' \times w' = 4l \times 4w = 16lw = 16A \). So area scales by \( k^2 \).

Step2: Analyze Original and New Area

From the table, original area seems \( 4.5 \) (units²), new area \( 18 \). Check \( 4.5 \times 16 = 72 \)? Wait, maybe original area calculation: Wait, maybe the original rectangle's area was \( 4.5 \), and with scale factor 4, area should be \( 4.5 \times 4^2 = 4.5 \times 16 = 72 \)? But the table has 18. Wait, maybe the scale factor is applied to linear dimensions, but maybe the original area was miscalculated. Wait, the question is how the scale factor changes the area. The key is: when scale factor \( k \) is applied to side lengths, area scales by \( k^2 \). So if scale factor is 4, area becomes \( 4^2 = 16 \) times original.

Step3: For Side Lengths (7b)

When a scale factor \( k = 4 \) is applied to a rectangle's side lengths (length and width), each side length is multiplied by 4. So new side lengths are 4 times original.

Answer:

7a) The area is multiplied by \( 4^2 = 16 \) (if original area is \( A \), new area is \( 16A \); from the table, \( 4.5 \times 16 = 72 \), but the table has 18—maybe a typo, but the concept is area scales by square of scale factor).
7b) Each side length is multiplied by 4 (new side lengths = 4 × original side lengths).

(Note: For 7a, using the scale factor effect: area scales with the square of the linear scale factor. So with scale factor 4, area changes by a factor of \( 4^2 = 16 \). If original area is \( 4.5 \), new area should be \( 4.5 \times 16 = 72 \), but the table has 18—possibly a mistake in the table, but the mathematical relationship is area scales by \( k^2 \). For 7b, linear dimensions scale by the scale factor \( k = 4 \), so side lengths are 4 times original.)