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Answer :
To determine the measure of the central angle associated with the arc Rob and his brother traveled, follow these steps:
1. Start with the information given:
- The diameter of the Ferris wheel is 40 feet.
- The distance traveled along the arc is [tex]\(\frac{86}{3} \pi\)[/tex] feet.
2. Calculate the radius of the Ferris wheel:
- Since the diameter is 40 feet, the radius is half of that:
[tex]\[
\text{Radius} = \frac{40}{2} = 20 \text{ feet}
\][/tex]
3. Use the formula to find the central angle in radians:
- The central angle in radians can be found using the formula:
[tex]\[
\text{Central Angle (radians)} = \frac{\text{Arc Length}}{\text{Radius}}
\][/tex]
- Substitute the values into the formula:
[tex]\[
\text{Central Angle (radians)} = \frac{\frac{86}{3} \pi}{20}
\][/tex]
4. Convert the central angle from radians to degrees:
- Radians can be converted to degrees by multiplying by [tex]\(\frac{180}{\pi}\)[/tex]:
[tex]\[
\text{Central Angle (degrees)} = \text{Central Angle (radians)} \times \frac{180}{\pi}
\][/tex]
5. Final result:
- After performing the calculations, we find that the central angle measures 258 degrees.
So, the measure of the associated central angle for the arc Rob and his brother traveled is [tex]\(258\)[/tex] degrees.
1. Start with the information given:
- The diameter of the Ferris wheel is 40 feet.
- The distance traveled along the arc is [tex]\(\frac{86}{3} \pi\)[/tex] feet.
2. Calculate the radius of the Ferris wheel:
- Since the diameter is 40 feet, the radius is half of that:
[tex]\[
\text{Radius} = \frac{40}{2} = 20 \text{ feet}
\][/tex]
3. Use the formula to find the central angle in radians:
- The central angle in radians can be found using the formula:
[tex]\[
\text{Central Angle (radians)} = \frac{\text{Arc Length}}{\text{Radius}}
\][/tex]
- Substitute the values into the formula:
[tex]\[
\text{Central Angle (radians)} = \frac{\frac{86}{3} \pi}{20}
\][/tex]
4. Convert the central angle from radians to degrees:
- Radians can be converted to degrees by multiplying by [tex]\(\frac{180}{\pi}\)[/tex]:
[tex]\[
\text{Central Angle (degrees)} = \text{Central Angle (radians)} \times \frac{180}{\pi}
\][/tex]
5. Final result:
- After performing the calculations, we find that the central angle measures 258 degrees.
So, the measure of the associated central angle for the arc Rob and his brother traveled is [tex]\(258\)[/tex] degrees.
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