The angle of elevation of the top of a tower from a point on the ground, which is 30 m away from the foot of the tower, is `30^(@)`. Find the height of the tower.

Answer

To find the height of the tower given the angle of elevation and the distance from the tower, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Right Triangle**:
- Let the height of the tower be \( AB \).
- The distance from the foot of the tower to the point on the ground is \( BC = 30 \) m.
- The angle of elevation from point B to the top of the tower (point A) is \( \theta = 30^\circ \).

2. **Use the Tangent Function**:
- In the right triangle \( ABC \), we can use the tangent function which relates the angle of elevation to the opposite side (height of the tower) and the adjacent side (distance from the tower).
- The formula is:
\[ \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} = \frac{AB}{BC} \]
- Substituting the known values:
\[ \tan(30^\circ) = \frac{AB}{30} \]

3. **Calculate \(\tan(30^\circ)\)**:
- We know that:
\[ \tan(30^\circ) = \frac{1}{\sqrt{3}} \]

4. **Set Up the Equation**:
- Substitute \(\tan(30^\circ)\) into the equation:
\[ \frac{1}{\sqrt{3}} = \frac{AB}{30} \]

5. **Solve for \(AB\)**:
- Cross-multiply to solve for \(AB\):
\[ AB = 30 \cdot \frac{1}{\sqrt{3}} = \frac{30}{\sqrt{3}} \]

6. **Rationalize the Denominator**:
- To simplify \(\frac{30}{\sqrt{3}}\), multiply the numerator and denominator by \(\sqrt{3}\):
\[ AB = \frac{30 \cdot \sqrt{3}}{3} = 10\sqrt{3} \]

7. **Approximate the Height**:
- Using the approximate value of \(\sqrt{3} \approx 1.73\):
\[ AB \approx 10 \cdot 1.73 = 17.3 \text{ m} \]

### Conclusion:
The height of the tower is approximately \( 17.3 \) meters.

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