Calculate the Final Leg: Distance in 1/4 Hour at 66 2/3 km/h

To solve this problem, we need to follow these steps:

• Step 1: Calculate the total time of the trip.
• Step 2: Use the average speed formula to find the total distance traveled.
• Step 3: Determine the distance traveled during the last $\frac{1}{4}$ hour by using known values.

Let's work through each step in detail:

Step 1: Calculate the total time of the trip.
The total time of the trip is the sum of all segments: riding, resting, and continuing.
Total time = $\frac{1}{3}$ hour riding + $\frac{1}{6}$ hour rest + $\frac{1}{4}$ hour riding = $\frac{2}{6} + \frac{1}{6} + \frac{1.5}{6} = \frac{4.5}{6} = \frac{3}{4}$ hour.

Step 2: Find the total distance using the given average speed.
Average speed formula: $\text{Average speed} = \frac{\text{Total distance}}{\text{Total time}}$.
Given that average speed is $66 \frac{2}{3} \, \text{km/h}$ or $\frac{200}{3} \, \text{km/h}$,
Total distance = Average speed $\times$ Total time
= $\frac{200}{3} \times \frac{3}{4} = 50 \, \text{km}$.

Step 3: Determine the distance covered in the last $\frac{1}{4}$ hour segment.
Subtract the known initial 30 km from the total distance of 50 km:
Distance covered in the last $\frac{1}{4}$ hour = Total distance - Distance in first segment
= $50 \, \text{km} - 30 \, \text{km} = 20 \, \text{km}$.

Therefore, the distance Rodney rides in the last quarter of an hour of his trip is $\textbf{20 km}$.