Step 1: Understanding the Concept:
Boat and stream problems are based on the concept of relative speed in a fluid medium.
Downstream: When a boat moves in the same direction as the current, the water's speed helps it, making it move faster (\(u + v\)).
Upstream: When moving against the current, the water's speed resists it, slowing it down (\(u - v\)).
By finding the speeds in both directions, we can calculate the boat's independent speed in still water (\(u\)) by taking the average of the two.
Step 2: Key Formula or Approach:
1. \(\text{Speed} = \text{Distance} / \text{Time}\).
2. \(\text{Still Water Speed (u)} = \frac{\text{Downstream Speed (D)} + \text{Upstream Speed (U)}}{2}\).
Step 3: Detailed Explanation:
First, calculate the speeds in both directions from the given data:
Downstream Speed (D): The boat covers 36 km in 3 hours.
\[ D = \frac{36}{3} = 12 \text{ km/h} \]
Upstream Speed (U): The boat covers 36 km in 4.5 hours.
\[ U = \frac{36}{4.5} = \frac{36 \times 2}{9} = 4 \times 2 = 8 \text{ km/h} \]
Now, use the still water speed formula to isolate the boat’s independent speed:
\[ u = \frac{D + U}{2} = \frac{12 + 8}{2} = \frac{20}{2} = 10 \text{ km/h} \]
Step 4: Final Answer:
The speed of the boat in still water is 10 km/h, which is option (c).