A person swims at 135° to current of river, to meet target on reaching opposite point. The ratio of person's velocity to river water velocity is

Options

A. `sqrt(3):1`
B. `sqrt(2):1`
C. `1:sqrt(2)`
D. `1:sqrt(3)`

Answer

To solve the problem of finding the ratio of the swimmer's velocity to the river's velocity when the swimmer swims at an angle of 135° to the current, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Angles**:
The swimmer swims at an angle of 135° with respect to the direction of the river current. This means that the swimmer's velocity vector forms a 135° angle with the river's velocity vector.

2. **Components of the Swimmer's Velocity**:
Let \( V_p \) be the swimmer's velocity and \( V_r \) be the river's velocity. We can break down the swimmer's velocity into its horizontal and vertical components:
- The horizontal component (against the current) is given by:
\[ V_{p_x} = V_p \cos(135°) \]
- The vertical component (across the river) is given by:
\[ V_{p_y} = V_p \sin(135°) \]

3. **Using Trigonometric Values**:
The cosine and sine of 135° can be calculated:
\[ \cos(135°) = -\frac{1}{\sqrt{2}}, \quad \sin(135°) = \frac{1}{\sqrt{2}} \]

4. **Setting Up the Equation**:
Since the swimmer aims to reach a point directly across the river, the horizontal component of the swimmer's velocity must equal the river's velocity:
\[ V_r = -V_p \cos(135°) \]
Substituting the value of \( \cos(135°) \):
\[ V_r = -V_p \left(-\frac{1}{\sqrt{2}}\right) = \frac{V_p}{\sqrt{2}} \]

5. **Finding the Ratio**:
To find the ratio of the swimmer's velocity to the river's velocity, we rearrange the equation:
\[ \frac{V_p}{V_r} = \sqrt{2} \]

6. **Final Answer**:
Thus, the ratio of the swimmer's velocity to the river's velocity is:
\[ \frac{V_p}{V_r} = \sqrt{2} : 1 \]