Determine the Distance a Boat Can Travel Against the Flow: A Step-by-Step Guide
In this article, you will learn about the process of solving a classic problem in river speed and distance calculation. We will walk through a detailed example and provide a step-by-step breakdown to understand how to determine the distance a boat can travel against the direction of a river's current. This guide is designed for educators, students, and anyone interested in understanding such problems through the lens of mathematical and logical reasoning.
The Problem at Hand
A boat travels 60 kilometers in 4 hours in the direction of the river's flow. Given that the speed of the boat is twice the speed of the river, we aim to determine how far the boat can travel in 2 hours going against the river's current.
Step-by-Step Calculation
1. **Calculate the Speed of the Boat with the Current
The boat covers 60 kilometers in 4 hours when moving with the current.
Speed of boat with the current frac{60 text{ km}}{4 text{ hours}} 15 text{ km/h}
2. **Define the Variables
Let the speed of the river be r km/h and the speed of the boat in still water be 2r km/h.
The effective speed of the boat with the current is given by:
Speed of boat with current Speed of boat Speed of river 2r r 3r
3. **Determine the Speed of the River
We know that:
3r 15 text{ km/h} implies text{ r} 5 text{ km/h}
4. **Calculate the Speed of the Boat in Still Water
The speed of the boat in still water is twice the speed of the river:
Speed of boat 2r 2 times 5 text{ km/h} 10 text{ km/h}
5. **Calculate the Effective Speed of the Boat Against the Current
The effective speed of the boat against the current is the speed of the boat in still water minus the speed of the river:
Speed of boat against current text{Speed of boat} - text{Speed of river} 10 text{ km/h} - 5 text{ km/h} 5 text{ km/h}
6. **Determine the Distance the Boat Can Travel in 2 Hours Against the Current
Distance Speed times Time
In 2 hours, the distance is:
Distance 5 text{ km/h} times 2 text{ hours} 10 text{ km}
Conclusion
Therefore, the boat can travel 10 kilometers in 2 hours against the direction of the river's flow.
Related Calculations
Starting from the initial conditions:
Let b and s be the speed of the boat and the speed of the river, respectively. Then s frac{60}{4} text{ km/h} 15 text{ km/h}
Given that b 2 s, we can solve:
b 2 times 15 30 text{ km/h}
The speed of the boat with the current is:
b - s 30 - 15 15 text{ km/h}
The speed of the river is:
s frac{15}{3} 5 text{ km/h}
The speed of the boat against the current is:
b - s 30 - 5 25 text{ km/h}
In 2 hours, the distance traveled is:
Distance 25 text{ km/h} times 2 text{ hours} 50 text{ km}
Therefore, in a different context, the boat can travel 50 kilometers in 2 hours against the flow of the river.