Welcome to Westonci.ca, where curiosity meets expertise. Ask any question and receive fast, accurate answers from our knowledgeable community. Explore a wealth of knowledge from professionals across various disciplines on our comprehensive Q&A platform. Our platform provides a seamless experience for finding reliable answers from a network of experienced professionals.
Sagot :
To solve for the measure of the associated central angle for the arc Rob and his brother traveled on the Ferris wheel, we follow these steps:
1. Diameter of the Ferris wheel:
The diameter of the Ferris wheel is given as 40 feet.
2. Distance traveled:
They traveled a distance of [tex]\(\frac{86}{3} \pi\)[/tex] feet.
3. Calculate the radius:
The radius [tex]\( r \)[/tex] is half of the diameter:
[tex]\[ r = \frac{40}{2} = 20 \text{ feet} \][/tex]
4. Find the circumference of the Ferris wheel:
The circumference [tex]\( C \)[/tex] is given by:
[tex]\[ C = 2 \pi r = 2 \pi \times 20 = 40 \pi \text{ feet} \][/tex]
5. Find the fraction of the circumference that they traveled:
The fraction of the circumference that equals the distance traveled is:
[tex]\[ \text{Fraction} = \frac{\text{Distance traveled}}{\text{Circumference}} = \frac{\frac{86}{3} \pi}{40 \pi} = \frac{86}{3 \times 40} = \frac{86}{120} = \frac{43}{60} \][/tex]
6. Calculate the central angle in radians:
Since one complete revolution equals [tex]\( 2\pi \)[/tex] radians, the central angle in radians is:
[tex]\[ \text{Central angle (radians)} = \text{Fraction} \times 2\pi = \frac{43}{60} \times 2\pi = \frac{86}{60} \pi = \frac{43}{30} \pi \approx 4.50294947014537 \text{ radians} \][/tex]
7. Convert the central angle to degrees:
To convert from radians to degrees, use the conversion factor [tex]\( 1 \text{ radian} = \frac{180}{\pi} \text{ degrees} \)[/tex]:
[tex]\[ \text{Central angle (degrees)} = 4.50294947014537 \text{ radians} \times \frac{180}{\pi} \approx 258.0 \text{ degrees} \][/tex]
So, the measure of the associated central angle for the arc they traveled is [tex]\( 258 \)[/tex] degrees.
Therefore, the central angle measures [tex]\( \boxed{258} \)[/tex].
1. Diameter of the Ferris wheel:
The diameter of the Ferris wheel is given as 40 feet.
2. Distance traveled:
They traveled a distance of [tex]\(\frac{86}{3} \pi\)[/tex] feet.
3. Calculate the radius:
The radius [tex]\( r \)[/tex] is half of the diameter:
[tex]\[ r = \frac{40}{2} = 20 \text{ feet} \][/tex]
4. Find the circumference of the Ferris wheel:
The circumference [tex]\( C \)[/tex] is given by:
[tex]\[ C = 2 \pi r = 2 \pi \times 20 = 40 \pi \text{ feet} \][/tex]
5. Find the fraction of the circumference that they traveled:
The fraction of the circumference that equals the distance traveled is:
[tex]\[ \text{Fraction} = \frac{\text{Distance traveled}}{\text{Circumference}} = \frac{\frac{86}{3} \pi}{40 \pi} = \frac{86}{3 \times 40} = \frac{86}{120} = \frac{43}{60} \][/tex]
6. Calculate the central angle in radians:
Since one complete revolution equals [tex]\( 2\pi \)[/tex] radians, the central angle in radians is:
[tex]\[ \text{Central angle (radians)} = \text{Fraction} \times 2\pi = \frac{43}{60} \times 2\pi = \frac{86}{60} \pi = \frac{43}{30} \pi \approx 4.50294947014537 \text{ radians} \][/tex]
7. Convert the central angle to degrees:
To convert from radians to degrees, use the conversion factor [tex]\( 1 \text{ radian} = \frac{180}{\pi} \text{ degrees} \)[/tex]:
[tex]\[ \text{Central angle (degrees)} = 4.50294947014537 \text{ radians} \times \frac{180}{\pi} \approx 258.0 \text{ degrees} \][/tex]
So, the measure of the associated central angle for the arc they traveled is [tex]\( 258 \)[/tex] degrees.
Therefore, the central angle measures [tex]\( \boxed{258} \)[/tex].
We appreciate your visit. Hopefully, the answers you found were beneficial. Don't hesitate to come back for more information. We appreciate your time. Please come back anytime for the latest information and answers to your questions. Thank you for visiting Westonci.ca, your go-to source for reliable answers. Come back soon for more expert insights.