The angle of elevation of a tower from a distance 200 m from its foot is `30^(@)`, Height of the tower is

Options

A. `100sqrt(3)m`
B. `(200)/(sqrt(3))m`
C. `200sqrt(3)m`
D. `(100)/(sqrt(3))m`

Answer

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

### Step-by-Step Solution:

1. **Identify the Given Information:**
- Distance from the foot of the tower (base) = 200 m
- Angle of elevation (θ) = 30°

2. **Draw a Right Triangle:**
- Let the height of the tower be represented as \( h \).
- In the right triangle formed, the opposite side is the height of the tower \( h \), and the adjacent side is the distance from the foot of the tower, which is 200 m.

3. **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:
\[ \tan(θ) = \frac{\text{opposite}}{\text{adjacent}} \]
- For our case:
\[ \tan(30°) = \frac{h}{200} \]

4. **Substitute the Value of \( \tan(30°) \):**
- We know that \( \tan(30°) = \frac{1}{\sqrt{3}} \).
- Substitute this value into the equation:
\[ \frac{1}{\sqrt{3}} = \frac{h}{200} \]

5. **Cross-Multiply to Solve for \( h \):**
- Cross-multiplying gives:
\[ h = 200 \times \frac{1}{\sqrt{3}} \]
- Simplifying this, we get:
\[ h = \frac{200}{\sqrt{3}} \text{ m} \]

6. **Final Answer:**
- The height of the tower is:
\[ h = \frac{200}{\sqrt{3}} \text{ m} \]