A river of width 80m is flowing at 6 m/s. A motorboat crosses the river in 10s and reaches a point directly across on the other river bank. The velocity of motorboat in still water is :

Options

A. 8 m/s
B. 10 m/s
C. 6 m/s
D. 5 m/s

Explanation

To solve the problem, we will use the concept of relative velocity and the Pythagorean theorem.

### Step-by-Step Solution:

1. **Identify the Given Values:**
- Width of the river (d) = 80 m
- Velocity of the river (Vr) = 6 m/s
- Time taken to cross the river (t) = 10 s

2. **Calculate the Velocity of the Motorboat Across the River:**
- The distance across the river is the width of the river.
- The speed of the motorboat in the direction perpendicular to the river (Vb_perpendicular) can be calculated using the formula:
\[ Vb_{\text{perpendicular}} = \frac{\text{Distance}}{\text{Time}} = \frac{80 \text{ m}}{10 \text{ s}} = 8 \text{ m/s} \]

3. **Set Up the Vector Relationship:**
- The velocity of the motorboat (Vb) can be broken down into two components:
- One component is perpendicular to the river (Vb_perpendicular = 8 m/s).
- The other component is parallel to the river (Vb_parallel = Vr = 6 m/s).
- Since these two components are perpendicular to each other, we can use the Pythagorean theorem to find the resultant velocity of the motorboat in still water (Vb_total):
\[ Vb_{\text{total}} = \sqrt{(Vb_{\text{perpendicular}})^2 + (Vb_{\text{parallel}})^2} \]
\[ Vb_{\text{total}} = \sqrt{(8 \text{ m/s})^2 + (6 \text{ m/s})^2} \]
\[ Vb_{\text{total}} = \sqrt{64 + 36} = \sqrt{100} = 10 \text{ m/s} \]

4. **Conclusion:**
- The velocity of the motorboat in still water is **10 m/s**.