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.
Similar Questions
- A boat covers a certain distance downstream in 2 hours, while it returns in $2\frac{1}{2}$ hours. If the speed of the current is 3 km/h, what is the speed of the boat in still water?
- A man can row a boat at 10km/h in still water. If the speed of the stream is 7km/h, what is the time taken to row a distance of 85 km down the stream?
- The speed of a boat in still water is 15 km/h and the speed of the current is one-third the speed of the boat in still water. How much time will it take to go 24 km upstream and 20 km downstream, assuming that no time is lost in changing direction?
- The speed of a boat in still water is 10 km/h and the speed of the stream is 3 km/h. How much time (in hours) will it take to sail 39 km downstream and then return immediately 28 km up stream?
- A boat can move 35 km upstream and the same distance downstream in a total time of 8 hours. If the speed of the boat in still water is 9 km/h, then the speed (in km/h) of the stream is:
- A man can row 9 km/h in still water. If the river is running at 4 km/h, it takes 8 hours more in upstream than to go downstream for the same distance. How far is the place?
- The speed of a boat in still water is 15 km/h and the speed of the current is 9 km/h. The distance travelled by the boat downstream in 25 minutes is:
- A motorboat, whose speed is 15 km/h in still water goes 20 km downstream and comes back in a total of 4 hours. The speed of the stream (in km/h) is:
- A boat covers a distance of 72 km downstream in 6 hours, while it takes 12 hours to cover the same distance upstream. What is the speed of the boat?
Important Questions from Boat and River
A boat sails 15 km of a river towards upstream in 5 hours. How long (in hours) will it take to cover the same distance downstream, if the speed of river is one-fourth the speed of the boat in still water?
The speed of boat upstream is 5 kmph. Its speed in still water is 10 kmph. How many minutes will it take to row 25 km downstream?
Sudha can travel a certain distance downstream in 6 hours by boat and return to the starting point in 9 hours. If the stream flows at a speed of 3 km/h, how long (in hours) will it take to cover a distance of 67.5 km in still water?
The downstream speed of a boat is 20 km/hr and the speed of stream is 4 km/hr. What will be the total time taken by the boat to cover 160 km downstream and 96 km upstream?
A man covered a distance of 18 km in 3 hours, 7.2 km in 2 hours and 15 km in 5 hours. What is the average distance traveled by him per hour?
