Boat Speed Calculation: Downstream and Upstream

This problem involves calculating the speed of a boat in still water when its speed is affected by the river's current. We are given the time taken for the boat to travel a certain distance downstream and the time taken to travel the same distance back upstream, along with the speed of the current.

Understanding the Concepts

To solve this, we need to understand how the boat's speed changes:

• Speed Downstream: When the boat travels downstream (with the current), the speed of the current adds to the boat's speed in still water. If $v_b$ is the speed of the boat in still water and $v_c$ is the speed of the current, the speed downstream ($v_d$) is $v_d = v_b + v_c$.
• Speed Upstream: When the boat travels upstream (against the current), the speed of the current subtracts from the boat's speed in still water. The speed upstream ($v_u$) is $v_u = v_b - v_c$.
• Distance, Speed, Time Relationship: The fundamental relationship is Distance = Speed × Time.

Given Information

• Time taken downstream ($t_d$): 2 hours
• Time taken upstream ($t_u$): $2\frac{1}{2}$ hours = 2.5 hours
• Speed of the current ($v_c$): 3 km/h
• Let the speed of the boat in still water be $v_b$ (in km/h).

Step-by-Step Solution

1. Define Speeds:
   • Speed downstream: $v_d = v_b + v_c = v_b + 3$ km/h
   • Speed upstream: $v_u = v_b - v_c = v_b - 3$ km/h
2. Calculate Distance: The distance covered downstream is the same as the distance covered upstream. Using the formula Distance = Speed × Time:
   • Distance downstream = $v_d \times t_d = (v_b + 3) \times 2$
   • Distance upstream = $v_u \times t_u = (v_b - 3) \times 2.5$
3. Equate Distances: Since the distance is the same for both journeys: $$ (v_b + 3) \times 2 = (v_b - 3) \times 2.5 $$
4. Solve for $v_b$: Now, we solve the equation for $v_b$.
   • Expand both sides: $2v_b + 6 = 2.5v_b - (3 \times 2.5)$
   • $2v_b + 6 = 2.5v_b - 7.5$
   • Rearrange the terms to group $v_b$ terms on one side and constants on the other: $6 + 7.5 = 2.5v_b - 2v_b$
   • $13.5 = 0.5v_b$
   • Isolate $v_b$: $v_b = \frac{13.5}{0.5}$
   • $v_b = 27$

Conclusion

The speed of the boat in still water ($v_b$) is 27 km/h. This matches one of the given options.