Two circles are concentric, and have radii of 15 and 25 respectively. If two arcs are formed by the same central angle, which of the following are possible arc lengths? Select the two correct answers.
30 and 40
50 and 80
3 and 5
45 and 75
60 and 90
the arcs will have the same ratio as the radii: 3:5
To find the arc length of a circle, we need to use the formula:
Arc length = (central angle / 360) x (2πr)
In this case, we have two concentric circles with radii of 15 and 25. Let's assume the central angle is θ.
For the circle with a radius of 15, the arc length will be (θ / 360) x (2π15).
For the circle with a radius of 25, the arc length will be (θ / 360) x (2π25).
Now, let's consider the given options:
1. 30 and 40 arc lengths: For option 1, if we divide 30 and 40 by (2π15 / 360), we should get an angle close to a whole number. Let's calculate:
- 30 / (2π15 / 360) ≈ 19.1 degrees (not a whole number)
- 40 / (2π15 / 360) ≈ 25.5 degrees (not a whole number)
Therefore, 30 and 40 are not possible arc lengths.
2. 50 and 80 arc lengths: For option 2, if we divide 50 and 80 by (2π15 / 360), we should get an angle close to a whole number. Let's calculate:
- 50 / (2π15 / 360) ≈ 31.8 degrees (not a whole number)
- 80 / (2π15 / 360) ≈ 50.9 degrees (not a whole number)
Therefore, 50 and 80 are not possible arc lengths.
3. 3 and 5 arc lengths: For option 3, if we divide 3 and 5 by (2π15 / 360), we should get an angle close to a whole number. Let's calculate:
- 3 / (2π15 / 360) ≈ 1.91 degrees (not a whole number)
- 5 / (2π15 / 360) ≈ 3.18 degrees (not a whole number)
Therefore, 3 and 5 are not possible arc lengths.
4. 45 and 75 arc lengths: For option 4, if we divide 45 and 75 by (2π15 / 360), we should get an angle close to a whole number. Let's calculate:
- 45 / (2π15 / 360) ≈ 28.6 degrees (not a whole number)
- 75 / (2π15 / 360) ≈ 47.7 degrees (not a whole number)
Therefore, 45 and 75 are not possible arc lengths.
5. 60 and 90 arc lengths: For option 5, if we divide 60 and 90 by (2π15 / 360), we should get an angle close to a whole number. Let's calculate:
- 60 / (2π15 / 360) = 38.2 degrees (approximately)
- 90 / (2π15 / 360) = 57.3 degrees (approximately)
Therefore, 60 and 90 are possible arc lengths.
Based on our calculations, the two correct answers are 60 and 90.