If a boat takes 4 h longer to travel a distance of 45 km upstream than to travel the same distance downstream, then find the speed of the boat in still water if the speed fo the stream is 2 km/h

Answer

To solve the problem, we need to find the speed of the boat in still water given that it takes 4 hours longer to travel 45 km upstream than downstream, and the speed of the stream is 2 km/h.

### Step-by-Step Solution:

1. **Define Variables:**
Let the speed of the boat in still water be \( x \) km/h.
The speed of the stream is given as \( y = 2 \) km/h.

2. **Calculate Effective Speeds:**
- **Upstream Speed:** When the boat is going upstream, its effective speed is \( x - y = x - 2 \) km/h.
- **Downstream Speed:** When the boat is going downstream, its effective speed is \( x + y = x + 2 \) km/h.

3. **Write Time Equations:**
- **Time taken upstream:** The time taken to travel 45 km upstream is given by:
\[ \text{Time upstream} = \frac{45}{x - 2} \]
- **Time taken downstream:** The time taken to travel 45 km downstream is given by:
\[ \text{Time downstream} = \frac{45}{x + 2} \]

4. **Set Up the Equation:**
According to the problem, the time taken upstream is 4 hours longer than the time taken downstream:
\[ \frac{45}{x - 2} = \frac{45}{x + 2} + 4 \]

5. **Clear the Fractions:**
Multiply through by \( (x - 2)(x + 2) \) to eliminate the denominators:
\[ 45(x + 2) = 45(x - 2) + 4(x - 2)(x + 2) \]

6. **Expand and Simplify:**
Expanding both sides gives:
\[ 45x + 90 = 45x - 90 + 4(x^2 - 4) \]
Simplifying further:
\[ 90 = -90 + 4x^2 - 16 \]
\[ 90 + 90 + 16 = 4x^2 \]
\[ 196 = 4x^2 \]
Dividing by 4:
\[ x^2 = 49 \]
Taking the square root:
\[ x = 7 \quad (\text{since speed cannot be negative}) \]

7. **Conclusion:**
The speed of the boat in still water is \( 7 \) km/h.