Suppose you consider a rectangular garden of 36 square meters with whole-number side lengths. What’s the dimension of design that has the least perimeter? What’s the dimension with the greatest perimeter?
Possible sizes:
4 * 9
3 * 12
2 * 18
P = 2L + 2W
P = (2 * 9) + (2 * 4)
P = ?
Find the perimeters of the other two sizes.
blah blah and blah
40
To find the dimension of the rectangular garden with the least perimeter, we need to determine the dimensions that will yield the smallest possible perimeter while still maintaining an area of 36 square meters.
Let's start by listing all possible sets of whole-number dimensions that result in an area of 36 square meters:
1. 1 × 36
2. 2 × 18
3. 3 × 12
4. 4 × 9
5. 6 × 6
6. 9 × 4
7. 12 × 3
8. 18 × 2
9. 36 × 1
Next, we calculate the perimeter for each set by adding the lengths of all four sides:
1. 1 × 36: Perimeter = 1 + 36 + 1 + 36 = 74
2. 2 × 18: Perimeter = 2 + 18 + 2 + 18 = 40
3. 3 × 12: Perimeter = 3 + 12 + 3 + 12 = 30
4. 4 × 9: Perimeter = 4 + 9 + 4 + 9 = 26
5. 6 × 6: Perimeter = 6 + 6 + 6 + 6 = 24
6. 9 × 4: Perimeter = 9 + 4 + 9 + 4 = 26
7. 12 × 3: Perimeter = 12 + 3 + 12 + 3 = 30
8. 18 × 2: Perimeter = 18 + 2 + 18 + 2 = 40
9. 36 × 1: Perimeter = 36 + 1 + 36 + 1 = 74
From the calculations, we can determine that the dimension with the least perimeter is 6 × 6, which has a perimeter of 24.
To find the dimension with the greatest perimeter, we can observe that as one side increases, the other decreases to maintain the same area. Therefore, the dimensions furthest from each other will result in the greatest perimeter.
The dimensions with the greatest perimeter are 1 × 36 and 36 × 1, both having a perimeter of 74.
In summary:
- The dimension with the least perimeter is 6 × 6 (perimeter = 24).
- The dimensions with the greatest perimeter are 1 × 36 and 36 × 1 (perimeter = 74).