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Answer :
Sure! Let's walk through the problem step-by-step to find the central angle measure.
1. Understand the given information:
- The diameter of the Ferris wheel is 40 feet.
- Rob and his brother traveled a distance of [tex]\(\frac{86}{3} \pi\)[/tex] feet along the arc before the ride stopped.
2. Calculate the radius of the Ferris wheel:
- The radius is half of the diameter:
[tex]\[
\text{radius} = \frac{\text{diameter}}{2} = \frac{40}{2} = 20 \text{ feet}
\][/tex]
3. Find the arc length in numerical form:
- The arc length traveled is given as [tex]\(\frac{86}{3} \pi\)[/tex] feet.
4. Use the formula for the arc length to find the central angle in radians:
- The formula for the arc length ([tex]\(s\)[/tex]) is given by:
[tex]\[
s = r \cdot \theta
\][/tex]
where [tex]\(r\)[/tex] is the radius, and [tex]\(\theta\)[/tex] is the central angle in radians.
- Rearrange the formula to solve for [tex]\(\theta\)[/tex] (the central angle):
[tex]\[
\theta = \frac{s}{r}
\][/tex]
- Substitute the given values into the formula:
[tex]\[
\theta = \frac{\frac{86}{3} \pi}{20}
\][/tex]
5. Simplifying calculations:
- First calculate [tex]\(\frac{86}{3}\)[/tex]:
[tex]\[
\frac{86}{3} \approx 28.6667
\][/tex]
- Substitute back in:
[tex]\[
\theta = \frac{28.6667 \cdot \pi}{20}
\][/tex]
- Divide by 20:
[tex]\[
\theta = 1.433333 \cdot \pi \approx 4.50294947014537 \text{ radians}
\][/tex]
6. Convert the central angle from radians to degrees:
- Use the conversion factor [tex]\(1 \text{ radian} = \frac{180}{\pi} \text{ degrees}\)[/tex]:
[tex]\[
\theta \text{ in degrees } = 4.50294947014537 \cdot \frac{180}{\pi}
\approx 258 \text{ degrees}
\][/tex]
Therefore, the measure of the associated central angle for the arc they traveled is [tex]\(258^\circ\)[/tex].
1. Understand the given information:
- The diameter of the Ferris wheel is 40 feet.
- Rob and his brother traveled a distance of [tex]\(\frac{86}{3} \pi\)[/tex] feet along the arc before the ride stopped.
2. Calculate the radius of the Ferris wheel:
- The radius is half of the diameter:
[tex]\[
\text{radius} = \frac{\text{diameter}}{2} = \frac{40}{2} = 20 \text{ feet}
\][/tex]
3. Find the arc length in numerical form:
- The arc length traveled is given as [tex]\(\frac{86}{3} \pi\)[/tex] feet.
4. Use the formula for the arc length to find the central angle in radians:
- The formula for the arc length ([tex]\(s\)[/tex]) is given by:
[tex]\[
s = r \cdot \theta
\][/tex]
where [tex]\(r\)[/tex] is the radius, and [tex]\(\theta\)[/tex] is the central angle in radians.
- Rearrange the formula to solve for [tex]\(\theta\)[/tex] (the central angle):
[tex]\[
\theta = \frac{s}{r}
\][/tex]
- Substitute the given values into the formula:
[tex]\[
\theta = \frac{\frac{86}{3} \pi}{20}
\][/tex]
5. Simplifying calculations:
- First calculate [tex]\(\frac{86}{3}\)[/tex]:
[tex]\[
\frac{86}{3} \approx 28.6667
\][/tex]
- Substitute back in:
[tex]\[
\theta = \frac{28.6667 \cdot \pi}{20}
\][/tex]
- Divide by 20:
[tex]\[
\theta = 1.433333 \cdot \pi \approx 4.50294947014537 \text{ radians}
\][/tex]
6. Convert the central angle from radians to degrees:
- Use the conversion factor [tex]\(1 \text{ radian} = \frac{180}{\pi} \text{ degrees}\)[/tex]:
[tex]\[
\theta \text{ in degrees } = 4.50294947014537 \cdot \frac{180}{\pi}
\approx 258 \text{ degrees}
\][/tex]
Therefore, the measure of the associated central angle for the arc they traveled is [tex]\(258^\circ\)[/tex].
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