A boat covers certain distance between two spots on a river taking `'t_(1)'` time, going down stream and `'t_(2)'` time goind upstream, what time will be taken by the boat to cover the same distance in still water:-

Options

A. Yes, `60^(@)`
B. Yes, `30^(@)`
C. No
D. Yes, `45^(@)`

Explanation

To solve the problem step by step, we can follow these instructions:

### Step 1: Define Variables
Let:
- \( d \) = distance between the two spots
- \( u \) = speed of the boat in still water
- \( v \) = speed of the river current
- \( t_1 \) = time taken to cover the distance downstream
- \( t_2 \) = time taken to cover the distance upstream

### Step 2: Write the Equations for Downstream and Upstream
When the boat is going downstream, the effective speed of the boat is the sum of its speed and the speed of the river:
\[ \text{Effective speed downstream} = u + v \]
Using the formula for speed, we have:
\[ d = (u + v) \cdot t_1 \quad \text{(1)} \]

When the boat is going upstream, the effective speed of the boat is the difference between its speed and the speed of the river:
\[ \text{Effective speed upstream} = u - v \]
Using the formula for speed again, we have:
\[ d = (u - v) \cdot t_2 \quad \text{(2)} \]

### Step 3: Rearranging the Equations
From equation (1):
\[ u + v = \frac{d}{t_1} \quad \text{(3)} \]
From equation (2):
\[ u - v = \frac{d}{t_2} \quad \text{(4)} \]

### Step 4: Solve the Equations
Now, we can add equations (3) and (4):
\[ (u + v) + (u - v) = \frac{d}{t_1} + \frac{d}{t_2} \]
This simplifies to:
\[ 2u = \frac{d}{t_1} + \frac{d}{t_2} \]
\[ 2u = d \left(\frac{1}{t_1} + \frac{1}{t_2}\right) \]
Thus, we can express \( u \):
\[ u = \frac{d}{2} \left(\frac{1}{t_1} + \frac{1}{t_2}\right) \quad \text{(5)} \]

### Step 5: Find the Time Taken in Still Water
The time taken to cover the same distance \( d \) in still water is given by:
\[ t = \frac{d}{u} \]
Substituting \( u \) from equation (5):
\[ t = \frac{d}{\frac{d}{2} \left(\frac{1}{t_1} + \frac{1}{t_2}\right)} \]
This simplifies to:
\[ t = \frac{2}{\left(\frac{1}{t_1} + \frac{1}{t_2}\right)} \]
\[ t = \frac{2 t_1 t_2}{t_1 + t_2} \]

### Final Answer
The time taken by the boat to cover the same distance in still water is:
\[ t = \frac{2 t_1 t_2}{t_1 + t_2} \]