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Answer :
To find the measure of the central angle associated with the arc that Rob and his brother traveled on the Ferris wheel, follow these steps:
1. Understand the Problem:
- The Ferris wheel has a diameter of 40 feet.
- The distance traveled (arc length) by Rob and his brother is [tex]\(\frac{86}{3} \pi\)[/tex] feet.
2. Find the Circumference of the Ferris Wheel:
- The formula for the circumference [tex]\(C\)[/tex] of a circle is:
[tex]\[
C = \pi \times \text{diameter}
\][/tex]
- Substitute the diameter of the Ferris wheel:
[tex]\[
C = \pi \times 40 = 40\pi
\][/tex]
3. Calculate the Central Angle in Radians:
- The formula to find the central angle [tex]\(\theta\)[/tex] in radians is:
[tex]\[
\theta = \frac{\text{arc length}}{\text{circumference}}
\][/tex]
- Substitute the given arc length and calculated circumference:
[tex]\[
\theta = \frac{\frac{86}{3} \pi}{40\pi} = \frac{86}{3 \times 40} = \frac{86}{120}
\][/tex]
- Simplify the fraction:
[tex]\[
\theta = \frac{43}{60} \approx 0.7167 \, \text{radians}
\][/tex]
4. Convert the Central Angle from Radians to Degrees:
- Use the conversion formula:
[tex]\[
\text{Degrees} = \text{Radians} \times \left(\frac{180}{\pi}\right)
\][/tex]
- Convert:
[tex]\[
\text{Degrees} = 0.7167 \times \left(\frac{180}{\pi}\right) \approx 41.06^\circ
\][/tex]
Therefore, the measure of the associated central angle for the arc traveled is approximately [tex]\(41.06^\circ\)[/tex].
1. Understand the Problem:
- The Ferris wheel has a diameter of 40 feet.
- The distance traveled (arc length) by Rob and his brother is [tex]\(\frac{86}{3} \pi\)[/tex] feet.
2. Find the Circumference of the Ferris Wheel:
- The formula for the circumference [tex]\(C\)[/tex] of a circle is:
[tex]\[
C = \pi \times \text{diameter}
\][/tex]
- Substitute the diameter of the Ferris wheel:
[tex]\[
C = \pi \times 40 = 40\pi
\][/tex]
3. Calculate the Central Angle in Radians:
- The formula to find the central angle [tex]\(\theta\)[/tex] in radians is:
[tex]\[
\theta = \frac{\text{arc length}}{\text{circumference}}
\][/tex]
- Substitute the given arc length and calculated circumference:
[tex]\[
\theta = \frac{\frac{86}{3} \pi}{40\pi} = \frac{86}{3 \times 40} = \frac{86}{120}
\][/tex]
- Simplify the fraction:
[tex]\[
\theta = \frac{43}{60} \approx 0.7167 \, \text{radians}
\][/tex]
4. Convert the Central Angle from Radians to Degrees:
- Use the conversion formula:
[tex]\[
\text{Degrees} = \text{Radians} \times \left(\frac{180}{\pi}\right)
\][/tex]
- Convert:
[tex]\[
\text{Degrees} = 0.7167 \times \left(\frac{180}{\pi}\right) \approx 41.06^\circ
\][/tex]
Therefore, the measure of the associated central angle for the arc traveled is approximately [tex]\(41.06^\circ\)[/tex].
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