A boat goes 30 km downstream and while returning covered only 80% of distance that covered in downstream . If boat takes 4 hr more to cover upstream than downstream then find the speed of boat in still water (km/hr) if speed of stream is `(25)/(36)` m/sec ?

Options

A. 8 km/hr
B. 2.8 km/hr
C. 5.6 km/hr
D. 4.2 km/hr

Explanation

To solve the problem step by step, we will first convert the speed of the stream from m/sec to km/hr, then set up the equations based on the information given, and finally solve for the speed of the boat in still water.

### Step-by-Step Solution:

1. **Convert Speed of Stream to km/hr**:
The speed of the stream is given as \( \frac{25}{36} \) m/sec. To convert this to km/hr:
\[ \text{Speed of stream} = \frac{25}{36} \times \frac{3600}{1000} = \frac{25 \times 10}{1} = 6.944 \text{ km/hr} \approx 2.5 \text{ km/hr} \]

**Hint**: Remember that to convert m/sec to km/hr, multiply by \( \frac{3600}{1000} \).

2. **Define Variables**:
Let the speed of the boat in still water be \( x \) km/hr.

3. **Determine Distances**:
The boat travels 30 km downstream and returns covering only 80% of that distance upstream:
\[ \text{Distance upstream} = 0.8 \times 30 = 24 \text{ km} \]

4. **Set Up Time Equations**:
The time taken to travel downstream and upstream can be expressed as:
- Downstream time:
\[ \text{Time downstream} = \frac{30}{x + 2.5} \]
- Upstream time:
\[ \text{Time upstream} = \frac{24}{x - 2.5} \]

5. **Set Up the Equation**:
According to the problem, the boat takes 4 hours more to cover upstream than downstream:
\[ \frac{24}{x - 2.5} - \frac{30}{x + 2.5} = 4 \]

6. **Clear the Fractions**:
Multiply through by \( (x - 2.5)(x + 2.5) \) to eliminate the denominators:
\[ 24(x + 2.5) - 30(x - 2.5) = 4(x^2 - 6.25) \]

7. **Expand and Simplify**:
Expanding both sides gives:
\[ 24x + 60 - 30x + 75 = 4x^2 - 25 \]
Combine like terms:
\[ -6x + 135 = 4x^2 - 25 \]
Rearranging gives:
\[ 4x^2 + 6x - 160 = 0 \]

8. **Solve the Quadratic Equation**:
Using the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \):
Here, \( a = 4, b = 6, c = -160 \):
\[ x = \frac{-6 \pm \sqrt{6^2 - 4 \cdot 4 \cdot (-160)}}{2 \cdot 4} \]
Calculate the discriminant:
\[ = \frac{-6 \pm \sqrt{36 + 2560}}{8} = \frac{-6 \pm \sqrt{2596}}{8} \]
Approximating gives:
\[ x \approx 5.6 \text{ km/hr} \]

9. **Conclusion**:
The speed of the boat in still water is approximately \( 5.6 \text{ km/hr} \).