1. The problem asks to find all rectangles with natural number dimensions whose area is 72 square units, and then find their perimeters. We also need to determine when the perimeter is biggest and least. 2. Next, we consider rectangles with a perimeter of 48 units and find how many such rectangles exist with natural number dimensions, and whether their areas vary. ### Step 1: Rectangles with area 72 - Let the length be $l$ and width be $w$, both natural numbers. - The area condition is $l \times w = 72$. - We find all pairs $(l,w)$ such that $l \times w = 72$. The factor pairs of 72 are: $$ (1,72), (2,36), (3,24), (4,18), (6,12), (8,9) $$ - For each pair, calculate the perimeter $P = 2(l + w)$: 1. $P = 2(1 + 72) = 2 \times 73 = 146$ 2. $P = 2(2 + 36) = 2 \times 38 = 76$ 3. $P = 2(3 + 24) = 2 \times 27 = 54$ 4. $P = 2(4 + 18) = 2 \times 22 = 44$ 5. $P = 2(6 + 12) = 2 \times 18 = 36$ 6. $P = 2(8 + 9) = 2 \times 17 = 34$ - The biggest perimeter is $146$ for the rectangle $1 \times 72$. - The least perimeter is $34$ for the rectangle $8 \times 9$. ### Step 2: Rectangles with perimeter 48 - Let length $l$ and width $w$ be natural numbers. - The perimeter condition is $2(l + w) = 48$ or $l + w = 24$. - We find all pairs $(l,w)$ with $l + w = 24$ and $l,w \in \mathbb{N}$. Possible pairs are: $$(1,23), (2,22), (3,21), (4,20), (5,19), (6,18), (7,17), (8,16), (9,15), (10,14), (11,13), (12,12)$$ - Number of such rectangles is 12. - Calculate the area for each: 1. $1 \times 23 = 23$ 2. $2 \times 22 = 44$ 3. $3 \times 21 = 63$ 4. $4 \times 20 = 80$ 5. $5 \times 19 = 95$ 6. $6 \times 18 = 108$ 7. $7 \times 17 = 119$ 8. $8 \times 16 = 128$ 9. $9 \times 15 = 135$ 10. $10 \times 14 = 140$ 11. $11 \times 13 = 143$ 12. $12 \times 12 = 144$ - The areas vary from 23 to 144. - The maximum area is $144$ for the square $12 \times 12$. ### Final answers: - For area 72, the biggest perimeter is 146 (1x72), the least perimeter is 34 (8x9). - For perimeter 48, there are 12 rectangles with varying areas from 23 to 144.