Sure, let's go through the detailed step-by-step solution to find the measure of the associated central angle for the arc Rob and his brother traveled on the Ferris wheel.
1. Determine the diameter of the Ferris wheel:
- The problem states the diameter is 40 feet.
2. Calculate the radius:
- The radius is half of the diameter.
- [tex]\[ \text{Radius} = \frac{\text{Diameter}}{2} = \frac{40}{2} = 20 \text{ feet} \][/tex]
3. Determine the arc length:
- The arc length they traveled is given as [tex]\( \frac{86}{3} \pi \text{ feet} \)[/tex].
- The numerical value of this arc length is approximately:
[tex]\[ \frac{86}{3} \pi \approx 90.0589894029074 \text{ feet} \][/tex]
4. Calculate the central angle in radians:
- The central angle in radians can be found using the formula:
[tex]\[ \text{Central angle in radians} = \frac{\text{Arc length}}{\text{Radius}} \][/tex]
- Plugging in the values:
[tex]\[ \text{Central angle in radians} = \frac{90.0589894029074}{20} \approx 4.50294947014537 \text{ radians} \][/tex]
5. Convert the central angle from radians to degrees:
- To convert from radians to degrees, use the conversion factor [tex]\( 180^\circ / \pi \)[/tex].
- Multiply the central angle in radians by this conversion factor:
[tex]\[ \text{Central angle in degrees} = 4.50294947014537 \times \frac{180}{\pi} \approx 258^\circ \][/tex]
Thus, the measure of the associated central angle for the arc they traveled is [tex]\( \boxed{258} \)[/tex] degrees.
