Determining the Speed of the Stream: A Boat’s Journey Upstream and Downstream

Imagine a boat navigating through the river, traveling both upstream and downstream. The boat travels 35 km upstream and 49 km downstream in 7 hours each time. How can we determine the speed of the stream? Let's explore this problem step-by-step.

The Problem and Given Data

A boat travels 35 km upstream and 49 km downstream in 7 hours each time. We need to find the speed of the stream. Let:

n-b speed of the boat in still water (km/h) n-s speed of the stream (km/h)

Understanding the Concepts of Speed, Distance, and Time

Let's first understand the relationship between speed, distance, and time:

Speed frac{Distance}{Time}

Formulating Equations

When the boat is traveling upstream, its effective speed is denoted by (b-s), and when traveling downstream, its effective speed is (b s). Given the distances and times:

Distance upstream 35 km Distance downstream 49 km Time taken upstream 7 hours Time taken downstream 7 hours

Setting Up the Equations

Upstream travel:

7 frac{35}{b-s} Rearranging gives:

b-s frac{35}{7} 5 quad text{(Equation 1)}

Downstream travel:

7 frac{49}{b s} Rearranging gives:

b s frac{49}{7} 7 quad text{(Equation 2)}

Solving the Equations

We now have a system of two linear equations:

b-s 5 b s 7

Adding these two equations:

2b 12 implies b 6 text{ km/h}

Substituting b 6 into Equation 1:

6-s 5 implies s 1 text{ km/h}

Conclusion

The speed of the stream is 1 text{ km/h}.

Alternative Approach

We can also solve the problem by considering the speed of the boat and the stream directly:

Let the speed of the boat be (x) km/h and the speed of the stream be (y) km/h. Upstream speed: (x-y frac{35}{7} 5 text{ km/h} Downstream speed: (x y frac{49}{7} 7 text{ km/h} Adding these two equations: (2x 12 implies x 6 text{ km/h}) Substituting (x 6) into (x-y 5): (6 - y 5 implies y 1 text{ km/h})

Real-World Context

In reality, the stream's speed may vary, and you would likely choose to travel in the slower parts of the river when going upstream and the faster parts when going downstream. This problem helps us understand the relative speeds and how to calculate them accurately.

This exploration of the problem provides a clear, step-by-step method for determining the speed of the stream using the principles of speed, distance, and time. Whether using algebraic equations or real-world intuition, the solution remains consistent and reliable.