A man rows a boat with a speed of 18 km/hr in northwest direction. The shoeline makes an angle of `15^(@)` south of west. Obtain the component of the velocity of the boat along the shoreline:

Options

A. 9 km/hr
B. `18(sqrt(3))/(2) km//hr`
C. `18 cos (15^(@)) km//hr`
D. `18 cos (75^(@)) km//hr`

Answer

To solve the problem, we need to find the component of the velocity of the boat along the shoreline. Here’s a step-by-step solution:

### Step 1: Understand the Directions
The boat is moving in the northwest direction, which is at an angle of 45 degrees from both the north and west axes. The shoreline is at an angle of 15 degrees south of west.

### Step 2: Determine the Angle Between the Boat's Velocity and the Shoreline
Since the shoreline is 15 degrees south of west, we can find the angle between the direction of the boat (northwest) and the shoreline.
- The northwest direction is 45 degrees from the west.
- The shoreline is 15 degrees south of west, which is equivalent to 360 - 15 = 345 degrees from the north.

Thus, the angle between the boat's direction and the shoreline is:
\[ \text{Angle} = 45^\circ + 15^\circ = 60^\circ \]

### Step 3: Calculate the Component of Velocity Along the Shoreline
The velocity of the boat is given as 18 km/hr. To find the component of this velocity along the shoreline, we can use the cosine of the angle we found:
\[ \text{Component of velocity along the shoreline} = V \cdot \cos(\theta) \]
Where:
- \( V = 18 \) km/hr (velocity of the boat)
- \( \theta = 60^\circ \) (angle between the boat's direction and the shoreline)

Now substituting the values:
\[ \text{Component} = 18 \cdot \cos(60^\circ) \]

### Step 4: Calculate \( \cos(60^\circ) \)
We know that:
\[ \cos(60^\circ) = \frac{1}{2} \]

### Step 5: Final Calculation
Now substituting back into the equation:
\[ \text{Component} = 18 \cdot \frac{1}{2} = 9 \text{ km/hr} \]

### Conclusion
The component of the velocity of the boat along the shoreline is **9 km/hr**.

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