a person aims at a bird on top of a 5 metre high pole with an elevation of `30^(@)` . If the bullet is fired, it will travel k metre before reaching the bird. The value of k (in meters ) is

Options

A. ` 5 sqrt(3) //2`
B. 10
C. `5 sqrt(3)`
D. `10 sqrt(3)`

Answer

To solve the problem, we need to find the distance \( k \) that the bullet travels to reach the bird on top of a 5-meter high pole, given that the angle of elevation is \( 30^\circ \).

### Step-by-Step Solution:

1. **Understand the Problem**:
We have a pole of height 5 meters and a person aiming at a bird on top of this pole with an angle of elevation of \( 30^\circ \). We need to find the distance \( k \) that the bullet travels (the hypotenuse of the triangle formed).

2. **Draw a Diagram**:
Visualize the scenario by drawing a right triangle where:
- The height of the pole (opposite side) = 5 meters
- The angle of elevation \( \theta = 30^\circ \)
- The distance \( k \) (hypotenuse) is what we need to find.

3. **Use the Sine Function**:
In a right triangle, the sine of an angle is defined as the ratio of the length of the opposite side to the length of the hypotenuse. Therefore, we can write:
\[ \sin(\theta) = \frac{\text{Opposite}}{\text{Hypotenuse}} = \frac{5}{k} \]
Here, \( \theta = 30^\circ \).

4. **Substitute the Value of Sine**:
We know that:
\[ \sin(30^\circ) = \frac{1}{2} \]
So, we can set up the equation:
\[ \frac{1}{2} = \frac{5}{k} \]

5. **Cross-Multiply to Solve for \( k \)**:
Cross-multiplying gives us:
\[ 1 \cdot k = 2 \cdot 5 \]
Therefore:
\[ k = 10 \]

6. **Conclusion**:
The distance \( k \) that the bullet travels before reaching the bird is \( 10 \) meters.

### Final Answer:
The value of \( k \) is \( 10 \) meters.

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