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Answer :
To solve this problem, we need to find the measure of the central angle associated with the arc Rob and his brother traveled on the Ferris wheel.
Let's break it down step-by-step.
1. Identify the given information:
- Diameter of the Ferris wheel: 40 feet
- Distance traveled: [tex]\(\frac{86}{3} \pi\)[/tex] feet
2. First, we need to find the radius of the Ferris wheel:
- Radius [tex]\( r \)[/tex] is half of the diameter.
- So, [tex]\( r = \frac{40}{2} = 20 \)[/tex] feet.
3. Use the formula for the arc length to find the central angle:
- The formula for arc length (s) is:
[tex]\[ s = r \cdot \theta \][/tex]
where [tex]\( \theta \)[/tex] is the central angle measured in radians and [tex]\( r \)[/tex] is the radius.
- Rearranging the formula to solve for [tex]\( \theta \)[/tex]:
[tex]\[ \theta = \frac{s}{r} \][/tex]
4. Substitute the given distance traveled and the radius into the formula:
- [tex]\( s = \frac{86}{3} \pi \)[/tex] feet
- [tex]\( r = 20 \)[/tex] feet
[tex]\[
\theta = \frac{\frac{86}{3} \pi}{20}
\][/tex]
5. Simplify the equation:
- First, simplify the fraction:
[tex]\[
\frac{\frac{86}{3} \pi}{20} = \frac{86 \pi}{3 \cdot 20} = \frac{86 \pi}{60} = \frac{43 \pi}{30}
\][/tex]
- So, the central angle in radians is:
[tex]\[
\theta = \frac{43 \pi}{30}
\][/tex]
6. Convert the central angle from radians to degrees:
- We know that [tex]\( \pi \)[/tex] radians is equivalent to 180 degrees.
- Therefore, to convert radians to degrees, we multiply by [tex]\(\frac{180}{\pi}\)[/tex]:
[tex]\[
\theta_{\text{degrees}} = \left(\frac{43 \pi}{30}\right) \cdot \left(\frac{180}{\pi}\right) = \left(\frac{43 \cdot 180}{30}\right) = 43 \times 6 = 258 \text{ degrees}
\][/tex]
Hence, the measure of the associated central angle for the arc they traveled is:
[tex]\[ \boxed{258} \bullet \][/tex]
So, the central angle measures [tex]\( \boxed{258} \)[/tex] degrees.
Let's break it down step-by-step.
1. Identify the given information:
- Diameter of the Ferris wheel: 40 feet
- Distance traveled: [tex]\(\frac{86}{3} \pi\)[/tex] feet
2. First, we need to find the radius of the Ferris wheel:
- Radius [tex]\( r \)[/tex] is half of the diameter.
- So, [tex]\( r = \frac{40}{2} = 20 \)[/tex] feet.
3. Use the formula for the arc length to find the central angle:
- The formula for arc length (s) is:
[tex]\[ s = r \cdot \theta \][/tex]
where [tex]\( \theta \)[/tex] is the central angle measured in radians and [tex]\( r \)[/tex] is the radius.
- Rearranging the formula to solve for [tex]\( \theta \)[/tex]:
[tex]\[ \theta = \frac{s}{r} \][/tex]
4. Substitute the given distance traveled and the radius into the formula:
- [tex]\( s = \frac{86}{3} \pi \)[/tex] feet
- [tex]\( r = 20 \)[/tex] feet
[tex]\[
\theta = \frac{\frac{86}{3} \pi}{20}
\][/tex]
5. Simplify the equation:
- First, simplify the fraction:
[tex]\[
\frac{\frac{86}{3} \pi}{20} = \frac{86 \pi}{3 \cdot 20} = \frac{86 \pi}{60} = \frac{43 \pi}{30}
\][/tex]
- So, the central angle in radians is:
[tex]\[
\theta = \frac{43 \pi}{30}
\][/tex]
6. Convert the central angle from radians to degrees:
- We know that [tex]\( \pi \)[/tex] radians is equivalent to 180 degrees.
- Therefore, to convert radians to degrees, we multiply by [tex]\(\frac{180}{\pi}\)[/tex]:
[tex]\[
\theta_{\text{degrees}} = \left(\frac{43 \pi}{30}\right) \cdot \left(\frac{180}{\pi}\right) = \left(\frac{43 \cdot 180}{30}\right) = 43 \times 6 = 258 \text{ degrees}
\][/tex]
Hence, the measure of the associated central angle for the arc they traveled is:
[tex]\[ \boxed{258} \bullet \][/tex]
So, the central angle measures [tex]\( \boxed{258} \)[/tex] degrees.
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