Introduction to the Boat Speed Problem

Understanding the relationship between the speed of a boat in still water and the speed of the stream is crucial for solving navigation and travel time problems. In this article, we will walk through the process of finding these two speeds using given travel data. This is a common problem analyzed in mathematics and is highly relevant for any scenario involving water travel.

Setting Up the Problem

Let's denote the speed of the boat in still water as b (in km/h) and the speed of the stream as s (in km/h). The following information is given:

The boat travels 30 km upstream and 28 km downstream in 7 hours. The boat travels 21 km upstream and returns in 5 hours.

In water travel, the speed of the boat upstream (against the current) is b - s, and the downstream (with the current) speed is b s.

Formulating the Equations

We can set up the following equations based on the given information:

First Scenario Equation

The time taken to travel upstream and downstream can be expressed as:

Time upstream 30 / (b - s)

Time downstream 28 / (b s)

The total time for the first scenario is 7 hours:

(frac{30}{b - s} frac{28}{b s} 7)

Second Scenario Equation

The time taken to travel 21 km upstream and return can be expressed as:

Time upstream 21 / (b - s)

Time downstream 21 / (b s)

The total time for the second scenario is 5 hours:

(frac{21}{b - s} frac{21}{b s} 5)

Solving the System of Equations

To solve the system of equations, we can use a substitution or elimination method. Let's proceed with the substitution method:

Substitution Method

Let x 1/(b - s) and y 1/(b s).

The equations can be rewritten as:

(32x 36y 7)

(4 48y 9)

Solving for x and y

Multiplying the first equation by 5/4:

(4x 45y frac{35}{4})

Subtracting this from the second equation:

(4 48y - (4x 45y) 9 - frac{35}{4})

(36y - 45y frac{36 - 35}{4})

(36y - 45y frac{1}{4})

(y frac{1}{12})

Substituting y 1/12 into the first equation:

(32x 36 cdot frac{1}{12} 7)

(32x 3 7)

(32x 4)

(x frac{1}{8})

Finally, solving for b and s:

(b - s frac{1}{x} 8)

(b s frac{1}{y} 12)

Adding the equations:

(2b 20)

(b 10)

Subtracting the equations:

(2s 4)

(s 2)

The final results are:

Speed of the boat in still water b 10 km/h

Speed of the stream s 2 km/h

Conclusion

In conclusion, the speed of the boat in still water is approximately 10 km/h, and the speed of the stream is approximately 2 km/h. This method demonstrates how to solve complex navigation problems using mathematical equations. By understanding these concepts, you can efficiently plan trips and ensure safe travel in various water conditions.