Solving the Speed of Boat in Still Water and Water Current

Are you struggling to find the speed of a boat in still water and the speed of the water current? This detailed guide will walk you through the process using clear and simple examples, making it easy to understand and apply to similar problems.

Understanding the Problem

We are given that a boat travels 20 km upstream in 6 hours and 18 km downstream in 4 hours. Our goal is to find the speed of the boat in still water (denoted as (b)) and the speed of the water current (denoted as (c)).

Step-by-Step Solution

To solve this problem, we will use the formulas for upstream and downstream travel. Here's a breakdown of the steps:

Step 1: Calculate the Speeds Upstream and Downstream

The effective speed of the boat when going upstream is b - c, and when going downstream is b c. Given the distances and times, we can calculate these speeds as follows:

Upstream:

Effective speed Distance / Time 20 km / 6 h 3.33 km/h
Therefore, (b - c 3.33)

Downstream:

Effective speed Distance / Time 18 km / 4 h 4.5 km/h
Therefore, (b c 4.5)

Step 2: Set Up the Equations

We now have a system of two equations:

Equation 1: (b - c 3.33) Equation 2: (b c 4.5)

Step 3: Solve the Equations

To eliminate (c), we add the two equations:

(b - c b c 3.33 4.5)

(2b 7.83)

(b frac{7.83}{2} 3.915 approx 3.92 text{ km/h})

Now, substitute (b) back into one of the equations to find (c). Using the second equation:

(4.5 - c 3.92)

(c 4.5 - 3.92 0.58 text{ km/h})

Final Result:
- Speed of the boat in still water ((b)) ≈ 3.92 km/h
- Speed of the water current ((c)) ≈ 0.58 km/h

Addressing Common Misconceptions

A common mistake or invalid result can occur with equations or logical errors. For instance, in the given problem, another solution is provided, but it yields an invalid result because the speed in downstream cannot be less than the speed in upstream.

Example: Another Approach

Let’s consider another approach using a similar problem where a boat travels 48 km upstream in 6 hours and 24 km downstream in 4 hours. The equations derived are:

Upstream: (x - y 8) (48/6 8 km/h) Downstream: (x y 6) (24/4 6 km/h)

Adding these equations:

(2x 14)

(x 7)

Substituting (x) in the first equation:

(7 - y 8)

(y -1)

Here, the speed of the current ((y)) is negative, which is not physically possible. This indicates that the given data or the problem setup might be incorrect.

Conclusion

Understanding the correct process and avoiding common mistakes is crucial when solving such problems. The correct answer for the provided problem is: - Speed of the boat in still water ≈ 3.92 km/h - Speed of the water current ≈ 0.58 km/h