Question
Find the scale factor and ratio of perimeters for a pair of similar octagons with areas 36 ft squared and 81 ft squared.
Answers
GPT 3.5
First, let's find the side length of each octagon using the formula for the area of a regular octagon:
Area = (2 + 2√2) * side^2
For the first octagon with an area of 36 ft squared:
36 = (2 + 2√2) * side^2
9 = (2 + 2√2) * side^2
9 / (2 + 2√2) = side^2
side = √(9 / (2 + 2√2))
side ≈ 2.828 ft
For the second octagon with an area of 81 ft squared:
81 = (2 + 2√2) * side^2
27 = (2 + 2√2) * side^2
27 / (2 + 2√2) = side^2
side = √(27 / (2 + 2√2))
side ≈ 4.24 ft
Now, let's find the scale factor by dividing the side length of the larger octagon by the side length of the smaller octagon:
4.24 / 2.828 ≈ 1.5
The scale factor between the two octagons is 1.5.
Finally, let's find the ratio of perimeters by multiplying the scale factor by the ratio of the side lengths:
1.5 * 8 = 12
Therefore, the ratio of perimeters between the two octagons is 12:8 or 3:2.
Area = (2 + 2√2) * side^2
For the first octagon with an area of 36 ft squared:
36 = (2 + 2√2) * side^2
9 = (2 + 2√2) * side^2
9 / (2 + 2√2) = side^2
side = √(9 / (2 + 2√2))
side ≈ 2.828 ft
For the second octagon with an area of 81 ft squared:
81 = (2 + 2√2) * side^2
27 = (2 + 2√2) * side^2
27 / (2 + 2√2) = side^2
side = √(27 / (2 + 2√2))
side ≈ 4.24 ft
Now, let's find the scale factor by dividing the side length of the larger octagon by the side length of the smaller octagon:
4.24 / 2.828 ≈ 1.5
The scale factor between the two octagons is 1.5.
Finally, let's find the ratio of perimeters by multiplying the scale factor by the ratio of the side lengths:
1.5 * 8 = 12
Therefore, the ratio of perimeters between the two octagons is 12:8 or 3:2.