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Answer :
To solve the problem, we need to find the measure of the central angle in degrees. Let's go through the steps:
1. Identify the Diameter of the Ferris Wheel:
- The diameter of the Ferris wheel is 40 feet.
2. Calculate the Radius:
- The radius (r) is half of the diameter. Hence, [tex]\( r = \frac{40}{2} = 20 \)[/tex] feet.
3. Determine the Circumference:
- The formula for the circumference (C) of a circle is [tex]\( C = 2 \pi r \)[/tex].
- Substituting the radius, [tex]\( C = 2 \pi \times 20 = 40 \pi \)[/tex] feet.
4. Distance Traveled:
- The distance Rob and his brother traveled is given as [tex]\( \frac{85}{3} \pi \)[/tex] feet.
5. Find the Proportion of the Circumference Traveled:
- The proportion of the circumference that they traveled is the distance traveled divided by the total circumference.
- [tex]\(\text{Proportion} = \frac{\frac{85}{3} \pi}{40 \pi} = \frac{85}{120} = \frac{17}{24}\)[/tex].
6. Convert the Proportion to a Central Angle:
- The Ferris wheel makes a full circle at 360°. Therefore, the central angle corresponding to the given arc length is calculated by multiplying the proportion by 360°.
- [tex]\( \text{Central angle} = \frac{17}{24} \times 360° = 255°\)[/tex].
Thus, the measure of the associated central angle is [tex]\( 255° \)[/tex].
1. Identify the Diameter of the Ferris Wheel:
- The diameter of the Ferris wheel is 40 feet.
2. Calculate the Radius:
- The radius (r) is half of the diameter. Hence, [tex]\( r = \frac{40}{2} = 20 \)[/tex] feet.
3. Determine the Circumference:
- The formula for the circumference (C) of a circle is [tex]\( C = 2 \pi r \)[/tex].
- Substituting the radius, [tex]\( C = 2 \pi \times 20 = 40 \pi \)[/tex] feet.
4. Distance Traveled:
- The distance Rob and his brother traveled is given as [tex]\( \frac{85}{3} \pi \)[/tex] feet.
5. Find the Proportion of the Circumference Traveled:
- The proportion of the circumference that they traveled is the distance traveled divided by the total circumference.
- [tex]\(\text{Proportion} = \frac{\frac{85}{3} \pi}{40 \pi} = \frac{85}{120} = \frac{17}{24}\)[/tex].
6. Convert the Proportion to a Central Angle:
- The Ferris wheel makes a full circle at 360°. Therefore, the central angle corresponding to the given arc length is calculated by multiplying the proportion by 360°.
- [tex]\( \text{Central angle} = \frac{17}{24} \times 360° = 255°\)[/tex].
Thus, the measure of the associated central angle is [tex]\( 255° \)[/tex].
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