Calculating the Speed of the Current

This problem involves understanding how the speed of the current affects the speed of a boat travelling downstream and upstream. When a boat travels downstream, the current helps it, so the boat's speed relative to the shore is the sum of its speed in still water and the speed of the current. When travelling upstream, the current opposes the boat, so the boat's speed relative to the shore is the difference between its speed in still water and the speed of the current.

Defining Variables and Formulas

Let's define the key variables:

• \( b \) = Speed of the boat in still water (in km/h)
• \( c \) = Speed of the current (in km/h)

Based on this, we can express the downstream and upstream speeds:

• Speed downstream = \( b + c \)
• Speed upstream = \( b - c \)

The general formula relating speed, distance, and time is:

\[ \text{Speed} = \frac{\text{Distance}}{\text{Time}} \]

Analyzing the Downstream Journey

We are given that the person rows 88 km downstream in 11 hours.

Using the speed formula:

Speed downstream \( = \frac{88 \text{ km}}{11 \text{ h}} = 8 \text{ km/h} \)

So, we can write our first equation:

\[ b + c = 8 \quad (\text{Equation 1}) \]

Analyzing the Upstream Journey

We are given that the person rows 72 km upstream in 12 hours.

Using the speed formula:

Speed upstream \( = \frac{72 \text{ km}}{12 \text{ h}} = 6 \text{ km/h} \)

So, we can write our second equation:

\[ b - c = 6 \quad (\text{Equation 2}) \]

Solving for the Speed of the Current

We now have a system of two linear equations with two variables (\( b \) and \( c \)):

1. \( b + c = 8 \)
2. \( b - c = 6 \)

To find the speed of the current (\( c \)), we can eliminate \( b \). The easiest way to do this is to subtract Equation 2 from Equation 1:

\[ (b + c) - (b - c) = 8 - 6 \]

Simplifying the left side:

\[ b + c - b + c = 2 \]

\[ 2c = 2 \]

Now, solve for \( c \):

\[ c = \frac{2}{2} \]

\[ c = 1 \text{ km/h} \]

The speed of the current is 1 km/h.

To find the speed of the boat in still water (\( b \)), we can substitute \( c = 1 \) into either equation. Using Equation 1:

\[ b + 1 = 8 \]

\[ b = 8 - 1 = 7 \text{ km/h} \]

So, the boat's speed in still water is 7 km/h, and the current speed is 1 km/h.

The question asks specifically for the speed of the current.

Conclusion

Based on the calculations, the speed of the current is 1 km/h.

Revision Table: Boat and Stream Concepts

Additional Information on Relative Speed in Water

Problems involving boats and streams are classic examples of relative speed. The speed of the boat is considered relative to the water it is moving in. The speed of the water (the current) is relative to the shore. When calculating speeds relative to the shore (which is what's usually meant by "speed" in these problems unless specified otherwise), we add or subtract the speed of the current.

• When going downstream, the current's velocity vector adds to the boat's velocity vector.
• When going upstream, the current's velocity vector opposes the boat's velocity vector.

This is why the downstream speed is always greater than the boat's speed in still water, and the upstream speed is always less than the boat's speed in still water. The difference between the boat's speed in still water and its upstream/downstream speed is exactly the speed of the current.

Understanding these relative speeds is crucial for solving problems involving boat movement in rivers or streams.