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].
