Find the scale factor and ratio of perimeters for a pair of similar octagons with areas 36 ft squared and 81 ft squared.
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.