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Answer :
To find the measure of the associated central angle for the arc traveled on the Ferris wheel, you can follow these steps:
1. Understand the Problem:
- You're given the diameter of the Ferris wheel, which is 40 feet.
- The arc length traveled by Rob and his brother is [tex]\(\frac{86}{3} \pi\)[/tex] feet.
2. Find the Radius:
- Since the diameter is 40 feet, the radius of the Ferris wheel is half of that.
[tex]\[
\text{Radius} = \frac{40}{2} = 20 \text{ feet}
\][/tex]
3. Use the Formula for Central Angle:
- The formula to find the central angle (in radians) is:
[tex]\[
\text{Central Angle (radians)} = \frac{\text{Arc Length}}{\text{Radius}}
\][/tex]
- Substitute the given values:
[tex]\[
\text{Central Angle (radians)} = \frac{\frac{86}{3} \pi}{20}
\][/tex]
4. Calculate the Central Angle:
- Compute the central angle in radians:
[tex]\[
\text{Central Angle (radians)} \approx 4.5029 \text{ radians}
\][/tex]
5. Convert Radians to Degrees:
- To convert the central angle from radians to degrees, use the conversion factor [tex]\(180^\circ/\pi\)[/tex].
[tex]\[
\text{Central Angle (degrees)} = 4.5029 \times \left(\frac{180}{\pi}\right)
\][/tex]
- Calculate the value:
[tex]\[
\text{Central Angle (degrees)} \approx 258.0^\circ
\][/tex]
Therefore, the measure of the associated central angle for the arc they traveled is approximately [tex]\(258.0^\circ\)[/tex].
1. Understand the Problem:
- You're given the diameter of the Ferris wheel, which is 40 feet.
- The arc length traveled by Rob and his brother is [tex]\(\frac{86}{3} \pi\)[/tex] feet.
2. Find the Radius:
- Since the diameter is 40 feet, the radius of the Ferris wheel is half of that.
[tex]\[
\text{Radius} = \frac{40}{2} = 20 \text{ feet}
\][/tex]
3. Use the Formula for Central Angle:
- The formula to find the central angle (in radians) is:
[tex]\[
\text{Central Angle (radians)} = \frac{\text{Arc Length}}{\text{Radius}}
\][/tex]
- Substitute the given values:
[tex]\[
\text{Central Angle (radians)} = \frac{\frac{86}{3} \pi}{20}
\][/tex]
4. Calculate the Central Angle:
- Compute the central angle in radians:
[tex]\[
\text{Central Angle (radians)} \approx 4.5029 \text{ radians}
\][/tex]
5. Convert Radians to Degrees:
- To convert the central angle from radians to degrees, use the conversion factor [tex]\(180^\circ/\pi\)[/tex].
[tex]\[
\text{Central Angle (degrees)} = 4.5029 \times \left(\frac{180}{\pi}\right)
\][/tex]
- Calculate the value:
[tex]\[
\text{Central Angle (degrees)} \approx 258.0^\circ
\][/tex]
Therefore, the measure of the associated central angle for the arc they traveled is approximately [tex]\(258.0^\circ\)[/tex].
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