If A boats takes 9 hours more to travel 65 km in upstream then to travel 60 km in downstream. If speed of boat in still water is `2(7)/(9)m//sec`, then find speed of stream in (km/hr).

Options

A. 7
B. 4
C. 8
D. 5

Answer

To solve the problem step by step, we will follow the logical flow of the given question regarding the boat's travel upstream and downstream.

### Step 1: Understand the given data
We know:
- The distance traveled upstream = 65 km
- The distance traveled downstream = 60 km
- The time taken upstream is 9 hours more than the time taken downstream.
- The speed of the boat in still water (x) = \(2 \frac{7}{9} \text{ m/s}\)

### Step 2: Convert the speed of the boat to km/hr
First, convert the speed of the boat from m/s to km/hr.
\[ x = 2 \frac{7}{9} = \frac{25}{9} \text{ m/s} \]
To convert m/s to km/hr, we use the conversion factor \( \frac{18}{5} \):
\[ x = \frac{25}{9} \times \frac{18}{5} = \frac{25 \times 18}{9 \times 5} = \frac{450}{45} = 10 \text{ km/hr} \]

### Step 3: Define the speed of the stream
Let the speed of the stream be \(y\) km/hr.

### Step 4: Determine the effective speeds
- Downstream speed = \(x + y = 10 + y\)
- Upstream speed = \(x - y = 10 - y\)

### Step 5: Set up the time equations
Using the formula for time \( \text{Time} = \frac{\text{Distance}}{\text{Speed}} \):
- Time taken to travel upstream:
\[ \text{Time}_{upstream} = \frac{65}{10 - y} \]
- Time taken to travel downstream:
\[ \text{Time}_{downstream} = \frac{60}{10 + y} \]

### Step 6: Set up the equation based on the time difference
According to the problem, the time taken upstream is 9 hours more than the time taken downstream:
\[ \frac{65}{10 - y} = \frac{60}{10 + y} + 9 \]

### Step 7: Solve the equation
Multiply through by the denominators to eliminate fractions:
\[ 65(10 + y) = 60(10 - y) + 9(10 - y)(10 + y) \]
Expanding both sides:
\[ 650 + 65y = 600 - 60y + 9(100 - y^2) \]
\[ 650 + 65y = 600 - 60y + 900 - 9y^2 \]
Combine like terms:
\[ 650 + 65y = 1500 - 60y - 9y^2 \]
Rearranging gives:
\[ 9y^2 + 125y - 850 = 0 \]

### Step 8: Use the quadratic formula
Using the quadratic formula \(y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\):
Here, \(a = 9\), \(b = 125\), and \(c = -850\):
\[ y = \frac{-125 \pm \sqrt{125^2 - 4 \cdot 9 \cdot (-850)}}{2 \cdot 9} \]
Calculating the discriminant:
\[ 125^2 = 15625 \]
\[ 4 \cdot 9 \cdot 850 = 30600 \]
So,
\[ y = \frac{-125 \pm \sqrt{15625 + 30600}}{18} \]
Calculating the square root:
\[ \sqrt{46225} = 215 \]
Thus,
\[ y = \frac{-125 + 215}{18} = \frac{90}{18} = 5 \]

### Step 9: Conclusion
The speed of the stream \(y\) is:
\[ \boxed{5 \text{ km/hr}} \]