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Answer :
To find the measure of the central angle associated with the distance Rob and his brother traveled on the Ferris wheel, we need to follow these steps:
1. Calculate the Circumference of the Ferris Wheel:
- The circumference [tex]\( C \)[/tex] of a circle can be found using the formula:
[tex]\[
C = \pi \times \text{diameter}
\][/tex]
- Given that the diameter of the Ferris wheel is 40 feet:
[tex]\[
C = \pi \times 40
\][/tex]
2. Determine the Fraction of the Wheel Traveled:
- Rob and his brother traveled a distance of [tex]\(\frac{86}{3} \pi\)[/tex] feet on the Ferris wheel. To find the fraction of the wheel they traveled, divide the distance traveled by the total circumference:
[tex]\[
\text{Fraction of Circle} = \frac{\frac{86}{3} \pi}{40\pi}
\][/tex]
- Simplifying the fraction, the [tex]\(\pi\)[/tex] terms cancel out, and you get:
[tex]\[
\text{Fraction of Circle} = \frac{86}{120} = \frac{43}{60}
\][/tex]
3. Calculate the Central Angle in Degrees:
- The full circle of a Ferris wheel is [tex]\(360^\circ\)[/tex]. Therefore, the central angle [tex]\(\theta\)[/tex] for the distance traveled is:
[tex]\[
\theta = 360^\circ \times \frac{43}{60}
\][/tex]
- When you perform the multiplication:
[tex]\[
\theta = 258^\circ
\][/tex]
The measure of the associated central angle for the arc they traveled is [tex]\(258^\circ\)[/tex].
1. Calculate the Circumference of the Ferris Wheel:
- The circumference [tex]\( C \)[/tex] of a circle can be found using the formula:
[tex]\[
C = \pi \times \text{diameter}
\][/tex]
- Given that the diameter of the Ferris wheel is 40 feet:
[tex]\[
C = \pi \times 40
\][/tex]
2. Determine the Fraction of the Wheel Traveled:
- Rob and his brother traveled a distance of [tex]\(\frac{86}{3} \pi\)[/tex] feet on the Ferris wheel. To find the fraction of the wheel they traveled, divide the distance traveled by the total circumference:
[tex]\[
\text{Fraction of Circle} = \frac{\frac{86}{3} \pi}{40\pi}
\][/tex]
- Simplifying the fraction, the [tex]\(\pi\)[/tex] terms cancel out, and you get:
[tex]\[
\text{Fraction of Circle} = \frac{86}{120} = \frac{43}{60}
\][/tex]
3. Calculate the Central Angle in Degrees:
- The full circle of a Ferris wheel is [tex]\(360^\circ\)[/tex]. Therefore, the central angle [tex]\(\theta\)[/tex] for the distance traveled is:
[tex]\[
\theta = 360^\circ \times \frac{43}{60}
\][/tex]
- When you perform the multiplication:
[tex]\[
\theta = 258^\circ
\][/tex]
The measure of the associated central angle for the arc they traveled is [tex]\(258^\circ\)[/tex].
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