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Answer :
To find the measure of the central angle for the arc traveled on the Ferris wheel, we need to follow these steps:
1. Determine the Circumference of the Ferris Wheel:
- The diameter of the Ferris wheel is given as 40 feet.
- The formula for the circumference of a circle is [tex]\( C = \pi \times \text{diameter} \)[/tex].
- So, the circumference of the Ferris wheel is [tex]\( \pi \times 40 = 40\pi \)[/tex] feet.
2. Find the Fraction of the Circumference That Rob and His Brother Traveled:
- The distance they traveled is given as [tex]\(\frac{86}{3} \pi\)[/tex] feet.
- To find the fraction of the total circumference they traveled, divide the distance traveled by the total circumference:
[tex]\[
\text{Fraction of circumference} = \frac{\frac{86}{3} \pi}{40 \pi} = \frac{86}{3 \times 40} = \frac{86}{120}
\][/tex]
- Simplifying [tex]\(\frac{86}{120}\)[/tex] gives [tex]\(\frac{43}{60}\)[/tex].
3. Calculate the Central Angle:
- A full circle is 360 degrees.
- The central angle of the arc they traveled is the same fraction of the full circle:
[tex]\[
\text{Central angle} = \frac{43}{60} \times 360
\][/tex]
- Simplifying this,
[tex]\[
\text{Central angle} = \frac{43 \times 360}{60} = 258 \text{ degrees}
\][/tex]
Therefore, the measure of the central angle for the arc they traveled is [tex]\( \boxed{258} \)[/tex] degrees.
1. Determine the Circumference of the Ferris Wheel:
- The diameter of the Ferris wheel is given as 40 feet.
- The formula for the circumference of a circle is [tex]\( C = \pi \times \text{diameter} \)[/tex].
- So, the circumference of the Ferris wheel is [tex]\( \pi \times 40 = 40\pi \)[/tex] feet.
2. Find the Fraction of the Circumference That Rob and His Brother Traveled:
- The distance they traveled is given as [tex]\(\frac{86}{3} \pi\)[/tex] feet.
- To find the fraction of the total circumference they traveled, divide the distance traveled by the total circumference:
[tex]\[
\text{Fraction of circumference} = \frac{\frac{86}{3} \pi}{40 \pi} = \frac{86}{3 \times 40} = \frac{86}{120}
\][/tex]
- Simplifying [tex]\(\frac{86}{120}\)[/tex] gives [tex]\(\frac{43}{60}\)[/tex].
3. Calculate the Central Angle:
- A full circle is 360 degrees.
- The central angle of the arc they traveled is the same fraction of the full circle:
[tex]\[
\text{Central angle} = \frac{43}{60} \times 360
\][/tex]
- Simplifying this,
[tex]\[
\text{Central angle} = \frac{43 \times 360}{60} = 258 \text{ degrees}
\][/tex]
Therefore, the measure of the central angle for the arc they traveled is [tex]\( \boxed{258} \)[/tex] degrees.
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