Question
What are the possible integer dimensions for a rectangular garden with a fixed perimeter of 50 meters?
Answers
GPT-4o mini
To find the possible integer dimensions for a rectangular garden with a fixed perimeter of 50 meters, we start with the formula for the perimeter of a rectangle:
\[
P = 2(l + w)
\]
where \( l \) is the length and \( w \) is the width. Given that the perimeter \( P \) is 50 meters, we can set up the equation:
\[
50 = 2(l + w)
\]
Dividing both sides by 2 gives:
\[
l + w = 25
\]
Next, we can express the width \( w \) in terms of the length \( l \):
\[
w = 25 - l
\]
Since both length and width must be positive integers, we establish the conditions:
1. \( l > 0 \)
2. \( w > 0 \)
From \( w = 25 - l \), we insist \( 25 - l > 0 \), which leads to \( l < 25 \).
Thus, \( l \) must satisfy:
\[
0 < l < 25
\]
Since \( l \) must also be an integer, the possible integer values for \( l \) range from 1 to 24. We now find the corresponding value of \( w \) for each:
- If \( l = 1 \), then \( w = 25 - 1 = 24 \) → Dimensions: \( (1, 24) \)
- If \( l = 2 \), then \( w = 25 - 2 = 23 \) → Dimensions: \( (2, 23) \)
- If \( l = 3 \), then \( w = 25 - 3 = 22 \) → Dimensions: \( (3, 22) \)
- If \( l = 4 \), then \( w = 25 - 4 = 21 \) → Dimensions: \( (4, 21) \)
- If \( l = 5 \), then \( w = 25 - 5 = 20 \) → Dimensions: \( (5, 20) \)
- If \( l = 6 \), then \( w = 25 - 6 = 19 \) → Dimensions: \( (6, 19) \)
- If \( l = 7 \), then \( w = 25 - 7 = 18 \) → Dimensions: \( (7, 18) \)
- If \( l = 8 \), then \( w = 25 - 8 = 17 \) → Dimensions: \( (8, 17) \)
- If \( l = 9 \), then \( w = 25 - 9 = 16 \) → Dimensions: \( (9, 16) \)
- If \( l = 10 \), then \( w = 25 - 10 = 15 \) → Dimensions: \( (10, 15) \)
- If \( l = 11 \), then \( w = 25 - 11 = 14 \) → Dimensions: \( (11, 14) \)
- If \( l = 12 \), then \( w = 25 - 12 = 13 \) → Dimensions: \( (12, 13) \)
- If \( l = 13 \), then \( w = 25 - 13 = 12 \) → Dimensions: \( (13, 12) \)
- If \( l = 14 \), then \( w = 25 - 14 = 11 \) → Dimensions: \( (14, 11) \)
- If \( l = 15 \), then \( w = 25 - 15 = 10 \) → Dimensions: \( (15, 10) \)
- If \( l = 16 \), then \( w = 25 - 16 = 9 \) → Dimensions: \( (16, 9) \)
- If \( l = 17 \), then \( w = 25 - 17 = 8 \) → Dimensions: \( (17, 8) \)
- If \( l = 18 \), then \( w = 25 - 18 = 7 \) → Dimensions: \( (18, 7) \)
- If \( l = 19 \), then \( w = 25 - 19 = 6 \) → Dimensions: \( (19, 6) \)
- If \( l = 20 \), then \( w = 25 - 20 = 5 \) → Dimensions: \( (20, 5) \)
- If \( l = 21 \), then \( w = 25 - 21 = 4 \) → Dimensions: \( (21, 4) \)
- If \( l = 22 \), then \( w = 25 - 22 = 3 \) → Dimensions: \( (22, 3) \)
- If \( l = 23 \), then \( w = 25 - 23 = 2 \) → Dimensions: \( (23, 2) \)
- If \( l = 24 \), then \( w = 25 - 24 = 1 \) → Dimensions: \( (24, 1) \)
Thus, the possible integer dimensions for the rectangular garden with a perimeter of 50 meters are:
\[
(1, 24), (2, 23), (3, 22), (4, 21), (5, 20), (6, 19), (7, 18), (8, 17), (9, 16), (10, 15), (11, 14), (12, 13), (13, 12), (14, 11), (15, 10), (16, 9), (17, 8), (18, 7), (19, 6), (20, 5), (21, 4), (22, 3), (23, 2), (24, 1)
\]
\[
P = 2(l + w)
\]
where \( l \) is the length and \( w \) is the width. Given that the perimeter \( P \) is 50 meters, we can set up the equation:
\[
50 = 2(l + w)
\]
Dividing both sides by 2 gives:
\[
l + w = 25
\]
Next, we can express the width \( w \) in terms of the length \( l \):
\[
w = 25 - l
\]
Since both length and width must be positive integers, we establish the conditions:
1. \( l > 0 \)
2. \( w > 0 \)
From \( w = 25 - l \), we insist \( 25 - l > 0 \), which leads to \( l < 25 \).
Thus, \( l \) must satisfy:
\[
0 < l < 25
\]
Since \( l \) must also be an integer, the possible integer values for \( l \) range from 1 to 24. We now find the corresponding value of \( w \) for each:
- If \( l = 1 \), then \( w = 25 - 1 = 24 \) → Dimensions: \( (1, 24) \)
- If \( l = 2 \), then \( w = 25 - 2 = 23 \) → Dimensions: \( (2, 23) \)
- If \( l = 3 \), then \( w = 25 - 3 = 22 \) → Dimensions: \( (3, 22) \)
- If \( l = 4 \), then \( w = 25 - 4 = 21 \) → Dimensions: \( (4, 21) \)
- If \( l = 5 \), then \( w = 25 - 5 = 20 \) → Dimensions: \( (5, 20) \)
- If \( l = 6 \), then \( w = 25 - 6 = 19 \) → Dimensions: \( (6, 19) \)
- If \( l = 7 \), then \( w = 25 - 7 = 18 \) → Dimensions: \( (7, 18) \)
- If \( l = 8 \), then \( w = 25 - 8 = 17 \) → Dimensions: \( (8, 17) \)
- If \( l = 9 \), then \( w = 25 - 9 = 16 \) → Dimensions: \( (9, 16) \)
- If \( l = 10 \), then \( w = 25 - 10 = 15 \) → Dimensions: \( (10, 15) \)
- If \( l = 11 \), then \( w = 25 - 11 = 14 \) → Dimensions: \( (11, 14) \)
- If \( l = 12 \), then \( w = 25 - 12 = 13 \) → Dimensions: \( (12, 13) \)
- If \( l = 13 \), then \( w = 25 - 13 = 12 \) → Dimensions: \( (13, 12) \)
- If \( l = 14 \), then \( w = 25 - 14 = 11 \) → Dimensions: \( (14, 11) \)
- If \( l = 15 \), then \( w = 25 - 15 = 10 \) → Dimensions: \( (15, 10) \)
- If \( l = 16 \), then \( w = 25 - 16 = 9 \) → Dimensions: \( (16, 9) \)
- If \( l = 17 \), then \( w = 25 - 17 = 8 \) → Dimensions: \( (17, 8) \)
- If \( l = 18 \), then \( w = 25 - 18 = 7 \) → Dimensions: \( (18, 7) \)
- If \( l = 19 \), then \( w = 25 - 19 = 6 \) → Dimensions: \( (19, 6) \)
- If \( l = 20 \), then \( w = 25 - 20 = 5 \) → Dimensions: \( (20, 5) \)
- If \( l = 21 \), then \( w = 25 - 21 = 4 \) → Dimensions: \( (21, 4) \)
- If \( l = 22 \), then \( w = 25 - 22 = 3 \) → Dimensions: \( (22, 3) \)
- If \( l = 23 \), then \( w = 25 - 23 = 2 \) → Dimensions: \( (23, 2) \)
- If \( l = 24 \), then \( w = 25 - 24 = 1 \) → Dimensions: \( (24, 1) \)
Thus, the possible integer dimensions for the rectangular garden with a perimeter of 50 meters are:
\[
(1, 24), (2, 23), (3, 22), (4, 21), (5, 20), (6, 19), (7, 18), (8, 17), (9, 16), (10, 15), (11, 14), (12, 13), (13, 12), (14, 11), (15, 10), (16, 9), (17, 8), (18, 7), (19, 6), (20, 5), (21, 4), (22, 3), (23, 2), (24, 1)
\]