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Answer :
To determine the measure of the central angle for the arc Rob and his brother traveled on the Ferris wheel, we can follow the steps outlined below:
1. Find the radius of the Ferris wheel:
- The diameter of the Ferris wheel is given as 40 feet.
- The radius is half of the diameter, so:
[tex]\[
\text{Radius} = \frac{40}{2} = 20 \text{ feet}
\][/tex]
2. Calculate the distance traveled:
- The distance traveled on the Ferris wheel is given as:
[tex]\[
\frac{86}{3} \pi \text{ feet}
\][/tex]
3. Derive the formula for the central angle in radians:
- The formula to calculate the central angle ([tex]\(\theta\)[/tex]) in radians, when you know the distance traveled along the circumference and the radius, is:
[tex]\[
\theta = \frac{\text{Distance Traveled}}{\text{Radius}}
\][/tex]
- Plugging in the values:
[tex]\[
\theta = \frac{\frac{86}{3} \pi}{20}
\][/tex]
- Simplifying this,
[tex]\[
\theta = \frac{86 \pi}{60} = \frac{43 \pi}{30} \text{ radians}
\][/tex]
4. Convert the angle from radians to degrees:
- To convert radians to degrees, use the conversion factor [tex]\(180^\circ / \pi\)[/tex]:
[tex]\[
\text{Angle in degrees} = \left( \frac{43 \pi}{30} \right) \times \frac{180}{\pi}
\][/tex]
- Simplifying this,
[tex]\[
\text{Angle in degrees} = \frac{43 \times 180}{30} = 258^\circ
\][/tex]
Thus, the measure of the associated central angle for the arc they traveled is:
[tex]\[
\boxed{258}
\][/tex]
1. Find the radius of the Ferris wheel:
- The diameter of the Ferris wheel is given as 40 feet.
- The radius is half of the diameter, so:
[tex]\[
\text{Radius} = \frac{40}{2} = 20 \text{ feet}
\][/tex]
2. Calculate the distance traveled:
- The distance traveled on the Ferris wheel is given as:
[tex]\[
\frac{86}{3} \pi \text{ feet}
\][/tex]
3. Derive the formula for the central angle in radians:
- The formula to calculate the central angle ([tex]\(\theta\)[/tex]) in radians, when you know the distance traveled along the circumference and the radius, is:
[tex]\[
\theta = \frac{\text{Distance Traveled}}{\text{Radius}}
\][/tex]
- Plugging in the values:
[tex]\[
\theta = \frac{\frac{86}{3} \pi}{20}
\][/tex]
- Simplifying this,
[tex]\[
\theta = \frac{86 \pi}{60} = \frac{43 \pi}{30} \text{ radians}
\][/tex]
4. Convert the angle from radians to degrees:
- To convert radians to degrees, use the conversion factor [tex]\(180^\circ / \pi\)[/tex]:
[tex]\[
\text{Angle in degrees} = \left( \frac{43 \pi}{30} \right) \times \frac{180}{\pi}
\][/tex]
- Simplifying this,
[tex]\[
\text{Angle in degrees} = \frac{43 \times 180}{30} = 258^\circ
\][/tex]
Thus, the measure of the associated central angle for the arc they traveled is:
[tex]\[
\boxed{258}
\][/tex]
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