A lamp post stands on a horizontal plane. From a point situated at a distance 150 m from its foot, the angle of elevation of the top is `30^(@).` What is the height of the lamp post?

Options

A. `50 m`
B. ` 50 sqrt3 m `
C. ` ( 50)/( sqrt3) m `
D. ` 100 m`

Answer

To find the height of the lamp post, we can use the concept of trigonometry, specifically the tangent function, which relates the angle of elevation to the opposite side (height of the lamp post) and the adjacent side (distance from the lamp post).

### Step-by-step Solution:

1. **Identify the triangle**:
- Let the height of the lamp post be \( AB \).
- The distance from the point to the foot of the lamp post is \( BC = 150 \) m.
- The angle of elevation from point C to the top of the lamp post (point A) is \( \angle ACB = 30^\circ \).

2. **Use the tangent function**:
- The tangent of an angle in a right triangle is defined as the ratio of the opposite side to the adjacent side. Therefore, we can write:
\[ \tan(30^\circ) = \frac{AB}{BC} \]

3. **Substitute the known values**:
- We know that \( BC = 150 \) m and \( \tan(30^\circ) = \frac{1}{\sqrt{3}} \). Substituting these values into the equation gives:
\[ \frac{1}{\sqrt{3}} = \frac{AB}{150} \]

4. **Cross-multiply to solve for AB**:
- Cross-multiplying gives:
\[ AB = 150 \cdot \frac{1}{\sqrt{3}} \]

5. **Simplify the expression**:
- To simplify \( AB \), we can multiply the numerator and denominator by \( \sqrt{3} \):
\[ AB = \frac{150 \cdot \sqrt{3}}{3} = 50\sqrt{3} \]

6. **Conclusion**:
- The height of the lamp post \( AB \) is \( 50\sqrt{3} \) meters.

### Final Answer:
The height of the lamp post is \( 50\sqrt{3} \) meters.

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