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Answer :
To find the measure of the central angle for the arc Rob and his brother traveled on the Ferris wheel, we can follow these steps:
1. Understand the Diameter and Circumference:
- The Ferris wheel has a diameter of 40 feet.
- The circumference of a circle (which is the complete distance around the Ferris wheel) is given by the formula:
[tex]\[
\text{Circumference} = \pi \times \text{diameter}
\][/tex]
- Using the given diameter, the circumference of the Ferris wheel is:
[tex]\[
\text{Circumference} = \pi \times 40 = 40\pi \text{ feet}
\][/tex]
2. Calculate the Central Angle:
- Rob and his brother traveled an arc distance of [tex]\(\frac{86}{3} \pi\)[/tex] feet.
- The central angle [tex]\(\theta\)[/tex] (in radians) corresponding to this arc length can be found using the formula:
[tex]\[
\theta = \frac{\text{arc length}}{\text{circumference}}
\][/tex]
- Substituting the values we have:
[tex]\[
\theta = \frac{\frac{86}{3} \pi}{40\pi} = \frac{86}{3 \times 40}
\][/tex]
- Simplifying this fraction will give the central angle in radians:
[tex]\[
\theta = \frac{86}{120} = \frac{43}{60}
\][/tex]
3. Convert Radians to Degrees:
- To convert the central angle from radians to degrees, we use the conversion factor [tex]\(180^\circ = \pi \text{ radians}\)[/tex].
- So, the central angle in degrees is:
[tex]\[
\text{Angle in degrees} = \left(\frac{43}{60}\right) \times \frac{180}{\pi^\circ}
\][/tex]
- Simplifying this gives the central angle in degrees as approximately 41.06 degrees.
Therefore, the measure of the associated central angle is approximately [tex]\(41.06\)[/tex] degrees.
1. Understand the Diameter and Circumference:
- The Ferris wheel has a diameter of 40 feet.
- The circumference of a circle (which is the complete distance around the Ferris wheel) is given by the formula:
[tex]\[
\text{Circumference} = \pi \times \text{diameter}
\][/tex]
- Using the given diameter, the circumference of the Ferris wheel is:
[tex]\[
\text{Circumference} = \pi \times 40 = 40\pi \text{ feet}
\][/tex]
2. Calculate the Central Angle:
- Rob and his brother traveled an arc distance of [tex]\(\frac{86}{3} \pi\)[/tex] feet.
- The central angle [tex]\(\theta\)[/tex] (in radians) corresponding to this arc length can be found using the formula:
[tex]\[
\theta = \frac{\text{arc length}}{\text{circumference}}
\][/tex]
- Substituting the values we have:
[tex]\[
\theta = \frac{\frac{86}{3} \pi}{40\pi} = \frac{86}{3 \times 40}
\][/tex]
- Simplifying this fraction will give the central angle in radians:
[tex]\[
\theta = \frac{86}{120} = \frac{43}{60}
\][/tex]
3. Convert Radians to Degrees:
- To convert the central angle from radians to degrees, we use the conversion factor [tex]\(180^\circ = \pi \text{ radians}\)[/tex].
- So, the central angle in degrees is:
[tex]\[
\text{Angle in degrees} = \left(\frac{43}{60}\right) \times \frac{180}{\pi^\circ}
\][/tex]
- Simplifying this gives the central angle in degrees as approximately 41.06 degrees.
Therefore, the measure of the associated central angle is approximately [tex]\(41.06\)[/tex] degrees.
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