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

Given:

1. Speed of the boat in still water: $v_b = 6$ km/hr.
2. Distance between A and B: $D = 20$ km.
3. Total time taken for the journey $A \to B \to A \to B$: $T = 10$ hours.

To Find:

• Speed of the stream $v_s$.

Steps to Solve:

1. Define the Speeds in Each Direction:

   • Downstream speed (from A to B):
     The speed of the boat downstream (with the current) is the sum of the boat's speed and the stream's speed: $v_{AB} = v_b + v_s = 6 + v_s \, \text{km/hr}$

   • Upstream speed (from B to A):
     The speed of the boat upstream (against the current) is the difference between the boat's speed and the stream's speed: $v_{BA} = v_b - v_s = 6 - v_s \, \text{km/hr}$

2. Calculate Time for Each Leg of the Journey:

   • Time from A to B (downstream):
     $t_{AB} = \frac{D}{v_{AB}} = \frac{20}{6 + v_s}$

   • Time from B to A (upstream):
     $t_{BA} = \frac{D}{v_{BA}} = \frac{20}{6 - v_s}$

   • Time for the second trip from A to B (downstream):
     This is the same as the first trip downstream: $t_{AB} = \frac{20}{6 + v_s}$

3. Set Up the Total Time Equation:

   The total time for the entire journey is the sum of all three times: $t_{AB} + t_{BA} + t_{AB} = 10$

   Substitute the expressions for $t_{AB}$ and $t_{BA}$: $\frac{20}{6 + v_s} + \frac{20}{6 - v_s} + \frac{20}{6 + v_s} = 10$

   Combine like terms: $2 \cdot \frac{20}{6 + v_s} + \frac{20}{6 - v_s} = 10$

   Simplify further: $\frac{40}{6 + v_s} + \frac{20}{6 - v_s} = 10$

4. Solve for $v_s$:

   To solve for $v_s$, clear the denominators by multiplying through by $(6 + v_s)(6 - v_s)$: $40(6 - v_s) + 20(6 + v_s) = 10(6 + v_s)(6 - v_s)$

   Expand both sides: $240 - 40v_s + 120 + 20v_s = 10(36 - v_s^2)$

   Combine like terms: $360 - 20v_s = 360 - 10v_s^2$

   Rearrange to form a quadratic equation: $10v_s^2 - 20v_s = 0$

   Factor out $10v_s$: $10v_s(v_s - 2) = 0$

   Set each factor to zero: $v_s = 0 \quad \text{or} \quad v_s = 2$

   Since the stream cannot have a speed of 0 (because it would mean no stream), we have: $v_s = 2 \, \text{km/hr}$

Answer:

The speed of the stream is 2 km/hr.

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

Here are some related questions for further exploration:

1. How would the total time change if the speed of the stream increased to 3 km/hr?
2. What would the total distance covered by the boat be in this journey?
3. If the speed of the boat in still water was 8 km/hr, how would the speed of the stream affect the total time?
4. Can you find the speed of the stream if the total journey time was 9 hours instead of 10?
5. What happens to the travel time if the stream is faster than the boat's speed in still water?

Tip: Always remember to check if your answers are physically meaningful, especially when solving quadratic equations!