A person observed that the angle of elevation at the top of a pole of height 5 metre is `30^(@)` . Then the distance of the person from the pole is :

Options

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

Answer

To solve the problem, we will use the concept of trigonometry, specifically the tangent function, which relates the angle of elevation to the height of the pole and the distance from the pole.

### Step-by-Step Solution:

1. **Identify the Elements**:
- Height of the pole (perpendicular) = 5 meters
- Angle of elevation (θ) = 30 degrees
- Distance from the person to the pole (base) = ?

2. **Use the Tangent Function**:
The tangent of an angle in a right triangle is defined as the ratio of the opposite side (height of the pole) to the adjacent side (distance from the person to the pole).
\[ \tan(\theta) = \frac{\text{Opposite}}{\text{Adjacent}} \]
Here, we can write:
\[ \tan(30^\circ) = \frac{5}{\text{Distance}} \]

3. **Find the Value of \(\tan(30^\circ)\)**:
From trigonometric tables or the unit circle, we know:
\[ \tan(30^\circ) = \frac{1}{\sqrt{3}} \]

4. **Set Up the Equation**:
Substitute the value of \(\tan(30^\circ)\) into the equation:
\[ \frac{1}{\sqrt{3}} = \frac{5}{\text{Distance}} \]

5. **Cross-Multiply to Solve for Distance**:
Cross-multiplying gives:
\[ \text{Distance} = 5 \cdot \sqrt{3} \]

6. **Calculate the Distance**:
The distance from the person to the pole is:
\[ \text{Distance} = 5\sqrt{3} \text{ meters} \]

### Final Answer:
The distance of the person from the pole is \(5\sqrt{3}\) meters.

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