Length of a river is 8 km. Speed of boat in still water is 6 km/h . boat takes 3 hours to go and come back. Find the speed of stream.

Answer

To solve the problem step by step, we will follow the given information and apply the relevant formulas.

### Step 1: Understand the given information
- Length of the river = 8 km
- Speed of the boat in still water (x) = 6 km/h
- Total time taken for the round trip = 3 hours

### Step 2: Define the variables
Let the speed of the stream be \( y \) km/h.

### Step 3: Calculate the effective speeds
- When the boat is going downstream (with the stream), its speed will be:
\[ \text{Speed downstream} = x + y = 6 + y \text{ km/h} \]
- When the boat is coming upstream (against the stream), its speed will be:
\[ \text{Speed upstream} = x - y = 6 - y \text{ km/h} \]

### Step 4: Set up the time equation
Using the formula for time, which is:
\[ \text{Time} = \frac{\text{Distance}}{\text{Speed}} \]
The time taken to go downstream is:
\[ \text{Time downstream} = \frac{8}{6 + y} \]
The time taken to come back upstream is:
\[ \text{Time upstream} = \frac{8}{6 - y} \]
The total time for the round trip is given as 3 hours:
\[ \frac{8}{6 + y} + \frac{8}{6 - y} = 3 \]

### Step 5: Solve the equation
First, we can simplify the equation:
\[ \frac{8}{6 + y} + \frac{8}{6 - y} = 3 \]
Multiply through by the common denominator \((6 + y)(6 - y)\):
\[ 8(6 - y) + 8(6 + y) = 3(6 + y)(6 - y) \]
This simplifies to:
\[ 48 - 8y + 48 + 8y = 3(36 - y^2) \]
Combining like terms gives:
\[ 96 = 108 - 3y^2 \]
Rearranging this gives:
\[ 3y^2 = 108 - 96 \]
\[ 3y^2 = 12 \]
\[ y^2 = 4 \]
Taking the square root gives:
\[ y = 2 \text{ km/h} \]

### Step 6: Conclusion
The speed of the stream is:
\[ \text{Speed of the stream} = 2 \text{ km/h} \]