Boat Speed Problem: Calculating Distance A to B
This problem involves calculating the distance between two points, A and B, given the speed of a boat in still water, the speed of the stream, and the total time taken for a round trip (going from A to B and back to A).
Problem Analysis
We are given the following information:
- Speed of the boat in still water ($v_b$): 6 km/h
- Speed of the stream ($v_s$): 1.5 km/h
- Total time for the round trip (A to B and B to A): 2 hours 40 minutes
We need to find the distance ($d$) between points A and B.
Key Concepts and Formulas
To solve this, we need to understand the concepts of upstream and downstream speeds:
- Downstream Speed: When the boat travels in the direction of the stream, its speed relative to the bank is the sum of the boat's speed and the stream's speed.
Downstream Speed = Speed of boat in still water + Speed of stream
$$ v_{down} = v_b + v_s $$ - Upstream Speed: When the boat travels against the direction of the stream, its speed relative to the bank is the difference between the boat's speed and the stream's speed.
Upstream Speed = Speed of boat in still water - Speed of stream
$$ v_{up} = v_b - v_s $$ - Time = Distance / Speed
Step-by-Step Solution
1. Calculate Upstream and Downstream Speeds
Using the given speeds:
- Downstream speed ($v_{down}$) = $6 \text{ km/h} + 1.5 \text{ km/h} = 7.5 \text{ km/h}$
- Upstream speed ($v_{up}$) = $6 \text{ km/h} - 1.5 \text{ km/h} = 4.5 \text{ km/h}$
2. Convert Total Time to Hours
The total time is given as 2 hours 40 minutes.
40 minutes = $\frac{40}{60}$ hours = $\frac{2}{3}$ hours.
Total time = $2 + \frac{2}{3}$ hours = $\frac{6}{3} + \frac{2}{3}$ hours = $\frac{8}{3}$ hours.
3. Set Up the Equation
Let the distance between A and B be $d$ km.
Time taken to travel downstream (A to B) = $\frac{\text{Distance}}{\text{Downstream Speed}} = \frac{d}{v_{down}} = \frac{d}{7.5}$ hours.
Time taken to travel upstream (B to A) = $\frac{\text{Distance}}{\text{Upstream Speed}} = \frac{d}{v_{up}} = \frac{d}{4.5}$ hours.
The total time for the round trip is the sum of the time taken for the downstream and upstream journeys:
$$ \frac{d}{7.5} + \frac{d}{4.5} = \frac{8}{3} $$4. Solve for Distance (d)
To solve the equation, we can first convert the decimal speeds to fractions:
- $7.5 = \frac{15}{2}$
- $4.5 = \frac{9}{2}$
Substitute these into the equation:
$$ \frac{d}{15/2} + \frac{d}{9/2} = \frac{8}{3} $$ $$ \frac{2d}{15} + \frac{2d}{9} = \frac{8}{3} $$Find a common denominator for 15 and 9, which is 45. Multiply the terms accordingly:
$$ \frac{2d \times 3}{15 \times 3} + \frac{2d \times 5}{9 \times 5} = \frac{8}{3} $$ $$ \frac{6d}{45} + \frac{10d}{45} = \frac{8}{3} $$ $$ \frac{16d}{45} = \frac{8}{3} $$Now, solve for $d$:
$$ d = \frac{8}{3} \times \frac{45}{16} $$Simplify the expression:
$$ d = \frac{8 \times 45}{3 \times 16} = \frac{1 \times 45}{3 \times 2} $$ $$ d = \frac{45}{6} $$ $$ d = \frac{15}{2} $$ $$ d = 7.5 \text{ km} $$Conclusion
The distance between points A and B is 7.5 km.
Important Questions from Boat and River
The speed of a ship in still water is 5 km/hr and the speed of the stream is 2 km/hr. Rohan rows to place at a distance of 21 km and comes back to the starting point. The total time taken by him is:
The speed of a boat in still water is 9 km/hr and the speed of stream is 3 km/hr. The difference between the upstream speed and downstream speed will be:
A boat can go 10 km upstream and 11 km downstream in a total time of 52 minutes, If the speed of the stream is 5 km/h, then what is the speed (in km/h) of the boat when going downstream?
The upstream speed of the boat is 40 km/hr and the speed of the boat in still water is 55 km/hr. What is the downstream speed of the boat?
A. 75 km/hr
B. 70 km/hr
C. 60 km/hr
D. 65 km/hrA boat moving upstream takes 8 hours 48 minutes to cover a distance while it takes 4 hours to return to the starting point, downstream. What is the ratio of the speed of boat in still water to that of water current?
