From a point 20 m away from the foot of a tower, the angle of elevation of the top of the tower is `30^(@)` . The height of the tower is

Options

A. ` 10 sqrt(3) m`
B. `20 sqrt(3)` m
C. `10/sqrt(3)` m
D. `(20)/(sqrt(3))m`

Answer

To find the height of the tower based on the given information, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Problem**: We have a right triangle formed by the tower, the distance from the point to the foot of the tower, and the line of sight to the top of the tower. The distance from the point to the foot of the tower is 20 m, and the angle of elevation to the top of the tower is 30 degrees.

2. **Label the Triangle**:
- Let the foot of the tower be point A.
- Let the top of the tower be point B.
- Let the point from where the angle is measured be point C.
- Therefore, AC = 20 m (the distance from point C to the foot of the tower A).
- Let the height of the tower (AB) be h.

3. **Use Trigonometric Ratios**: In right triangle ABC, we can use the tangent function, which is defined as:
\[ \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} \]
Here, the angle of elevation θ is 30 degrees, the opposite side is the height of the tower (h), and the adjacent side is the distance from the point to the foot of the tower (20 m).

4. **Set Up the Equation**:
\[ \tan(30^\circ) = \frac{h}{20} \]

5. **Find the Value of tan(30°)**:
\[ \tan(30^\circ) = \frac{1}{\sqrt{3}} \]

6. **Substitute the Value into the Equation**:
\[ \frac{1}{\sqrt{3}} = \frac{h}{20} \]

7. **Solve for h**:
\[ h = 20 \times \frac{1}{\sqrt{3}} = \frac{20}{\sqrt{3}} \text{ meters} \]

8. **Rationalize the Denominator (if necessary)**:
\[ h = \frac{20\sqrt{3}}{3} \text{ meters} \]

### Final Answer:
The height of the tower is \( \frac{20}{\sqrt{3}} \) meters or approximately \( 11.55 \) meters.