The speed of a boat in still water is 9 km/h. Length of the river is 36 km. The boat takes 9 hours to go and come back. Find the speed of the stream.

Let's solve the problem step-by-step.

### Given:
1. Speed of the boat in still water: \( v_b = 9 \) km/hr.
2. Distance of the river (one way): \( D = 36 \) km.
3. Total time taken for the journey (going and coming back): \( T = 9 \) hours.

### To Find:
- Speed of the stream \( v_s \).

### Steps to Solve:

1. **Define the Speeds in Each Direction:**

   - **Downstream speed (going with the stream):**  
   The speed of the boat downstream is the sum of the boat's speed in still water and the stream's speed:
   \[
   v_{\text{down}} = v_b + v_s = 9 + v_s \, \text{km/hr}
   \]

   - **Upstream speed (going against the stream):**  
   The speed of the boat upstream is the difference between the boat's speed in still water and the stream's speed:
   \[
   v_{\text{up}} = v_b - v_s = 9 - v_s \, \text{km/hr}
   \]

2. **Calculate Time for Each Leg of the Journey:**

   - **Time taken downstream (from start to end of river):**  
   \[
   t_{\text{down}} = \frac{D}{v_{\text{down}}} = \frac{36}{9 + v_s}
   \]

   - **Time taken upstream (from end back to start):**  
   \[
   t_{\text{up}} = \frac{D}{v_{\text{up}}} = \frac{36}{9 - v_s}
   \]

3. **Set Up the Total Time Equation:**

   The total time for the journey (going downstream and coming back upstream) is the sum of both times:
   \[
   t_{\text{down}} + t_{\text{up}} = T
   \]
   Substituting the expressions for \( t_{\text{down}} \) and \( t_{\text{up}} \):
   \[
   \frac{36}{9 + v_s} + \frac{36}{9 - v_s} = 9
   \]

4. **Solve for \( v_s \):**

   First, divide the entire equation by 9 to simplify:
   \[
   \frac{4}{9 + v_s} + \frac{4}{9 - v_s} = 1
   \]

   Clear the denominators by multiplying through by \((9 + v_s)(9 - v_s)\):
   \[
   4(9 - v_s) + 4(9 + v_s) = (9 + v_s)(9 - v_s)
   \]

   Expand both sides:
   \[
   36 - 4v_s + 36 + 4v_s = 81 - v_s^2
   \]

   Combine like terms:
   \[
   72 = 81 - v_s^2
   \]

   Rearrange to isolate \( v_s^2 \):
   \[
   v_s^2 = 81 - 72
   \]
   \[
   v_s^2 = 9
   \]

   Take the square root of both sides:
   \[
   v_s = \sqrt{9} = 3
   \]

### Answer:
The speed of the stream is **3 km/hr**.

Would you like more details or have any questions about this solution?

Here are some related questions for further exploration:

1. What would be the total time taken if the speed of the stream increased to 4 km/hr?
2. How would the total journey time change if the river length was doubled to 72 km?
3. If the boat took 10 hours instead of 9 hours for the same journey, what would the stream's speed be?
4. Can you determine the speed of the boat in still water if the stream's speed was 5 km/hr and the total time was 8 hours?
5. What happens to the upstream time if the stream's speed equals the boat's speed in still water?

**Tip:** When solving problems involving relative motion in water, always distinguish between upstream and downstream conditions to avoid confusion.

Learn how to calculate the speed of a river stream using the boat's journey time and river length. Detailed step-by-step solution with formulas for time, boat speed, and stream speed. Suitable for advanced high school students and anyone interested in solving river navigation problems.

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