## Answer :

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.