A runner is jogging in a straight line at a
steady vr= 8.5 km/hr. When the runner is
L= 6.4 km from the finish line, a bird begins
flying straight from the runner to the finish
line at vb= 17 km/hr (2 times as fast as the
runner). When the bird reaches the finish
line, it turns around and flies directly back to
the runner.
L
vb
vr
finish
line
What cumulative distance does the bird
travel? Even though the bird is a dodo, assume that it occupies only one point in space
(a “zero” length bird), travels in a straight
line, and that it can turn without loss of
speed.
Answer in units of km.

To solve this problem, we need to calculate the total distance traveled by the bird before it meets the runner. The bird is continuously flying between the runner and the finish line, while the runner is jogging toward the finish line at a steady speed.

### Given:
- Speed of the runner: \( v_r = 8.5 \, \text{km/h} \)
- Speed of the bird: \( v_b = 17 \, \text{km/h} \) (twice the speed of the runner)
- Initial distance between the runner and the finish line: \( L = 6.4 \, \text{km} \)

### Step-by-step solution:

1. **Time for the runner to reach the finish line:**
   The time it will take for the runner to reach the finish line is calculated as:
   \[
   t_{\text{runner}} = \frac{L}{v_r} = \frac{6.4}{8.5} \, \text{hours}.
   \]

2. **Total time available for the bird to fly:**
   Since the bird flies back and forth between the runner and the finish line, the total time the bird is flying is the same as the time it takes the runner to reach the finish line:
   \[
   t_{\text{bird}} = t_{\text{runner}}.
   \]

3. **Total distance traveled by the bird:**
   The bird flies continuously for the same duration but at a speed of \( v_b = 17 \, \text{km/h} \). The total distance the bird travels is:
   \[
   D_{\text{bird}} = v_b \times t_{\text{bird}} = 17 \times t_{\text{runner}}.
   \]

4. **Substitute \( t_{\text{runner}} \) from Step 1 into this equation:**
   \[
   D_{\text{bird}} = 17 \times \frac{6.4}{8.5}.
   \]

Now let's compute the final value:

\[
D_{\text{bird}} = 17 \times \frac{6.4}{8.5} = 12.8 \, \text{km}.
\]

### Conclusion:
The bird travels a cumulative distance of \( \boxed{12.8} \, \text{km} \).

---

Would you like more details or have any questions?

Here are 5 related questions to consider:
1. How would the solution change if the bird's speed was less than twice the runner's speed?
2. How much time does the bird spend flying back and forth?
3. What happens if the runner's speed increases while the bird is in the air?
4. How would we calculate the time it takes for the bird to reach the finish line on its first trip?
5. Can we generalize this problem for different bird and runner speeds?

**Tip**: Always double-check units for consistency, especially when converting between different measures like km and hours.

A runner is jogging in a straight line at a steady vr= 8.5 km/hr. When the runner is L= 6.4 km from the finish line, a bird begins flying straight from the runner to the finish line at vb= 17 km/hr (2 times as fast as the runner). When the bird reaches the finish line, it turns around and flies directly back to the runner. What cumulative distance does the bird travel? Even though the bird is a dodo, assume that it occupies only one point in space (a 'zero' length bird), travels in a straight line, and that it can turn without loss of speed. Answer in units of km.

Solve the problem of calculating the cumulative distance a bird travels as it flies between a runner and the finish line. This involves speed-time relationships and distance calculations using algebraic principles. Suitable for students in Grades 9-12.

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