Sure, let's solve this step-by-step.
1. Identify Known Variables:
- Time taken to travel downstream: [tex]\( t_d = 5 \)[/tex] hours
- Time taken to travel upstream: [tex]\( t_u = 7 \)[/tex] hours
- Speed of the stream: [tex]\( s = 3 \)[/tex] km/hr
2. Define the Required Variable:
- Speed of the streamer in still water: [tex]\( v \)[/tex] km/hr
3. Equation for Downstream:
- When the boat is going downstream, the speed of the boat relative to the ground is the sum of the speed of the boat in still water and the speed of the stream.
- Downstream speed ([tex]\( v_d \)[/tex]) = Speed in still water ([tex]\( v \)[/tex]) + Speed of stream ([tex]\( s \)[/tex])
- Distance traveled downstream = Downstream speed [tex]\(\times\)[/tex] time = [tex]\( (v + 3) \times 5 \)[/tex]
4. Equation for Upstream:
- When the boat is going upstream, the speed of the boat relative to the ground is the difference between the speed of the boat in still water and the speed of the stream.
- Upstream speed ([tex]\( v_u \)[/tex]) = Speed in still water ([tex]\( v \)[/tex]) - Speed of stream ([tex]\( s \)[/tex])
- Distance traveled upstream = Upstream speed [tex]\(\times\)[/tex] time = [tex]\( (v - 3) \times 7 \)[/tex]
5. Equating the Distances:
- Since the distance between the two ports is the same both upstream and downstream, we can set the downstream and upstream distance equations equal to each other.
[tex]\[
(v + 3) \times 5 = (v - 3) \times 7
\][/tex]
6. Solving the Equation:
- Expand both sides of the equation:
[tex]\[
5v + 15 = 7v - 21
\][/tex]
- Combine like terms:
[tex]\[
15 + 21 = 7v - 5v
\][/tex]
[tex]\[
36 = 2v
\][/tex]
- Divide both sides by 2:
[tex]\[
v = \frac{36}{2} = 18
\][/tex]
7. Conclusion:
- The speed of the streamer in still water is [tex]\( 18 \)[/tex] km/hr.
