The angle of elevation of the top of a tower. from a point on the ground and at a distance of 160 m from its foot, is find to be `60^(@)` . Find the height of the tower .

Options

A. `277.12` m
B. `377.12` m
C. `477.12` m
D. None

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. **Draw a Diagram**:
- Let the tower be represented as line segment PQ, where P is the top of the tower and Q is the base of the tower.
- Let point A be the point on the ground from where the angle of elevation is measured.
- The distance from point A to the foot of the tower (point Q) is given as 160 meters.

2. **Identify the Angle of Elevation**:
- The angle of elevation from point A to the top of the tower (point P) is given as \(60^\circ\).

3. **Set Up the Right Triangle**:
- In triangle PQA, we have:
- PQ as the height of the tower (which we need to find).
- AQ as the distance from point A to the base of the tower, which is 160 m.
- Angle PAQ (angle of elevation) is \(60^\circ\).

4. **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 tower, PQ) to the adjacent side (distance from the tower, AQ).
- Therefore, we can write:
\[ \tan(60^\circ) = \frac{PQ}{AQ} \]
- Substituting the known values:
\[ \tan(60^\circ) = \frac{H}{160} \]

5. **Substitute the Value of \(\tan(60^\circ)\)**:
- We know that \(\tan(60^\circ) = \sqrt{3} \approx 1.732\).
- Thus, the equation becomes:
\[ \sqrt{3} = \frac{H}{160} \]

6. **Solve for H (Height of the Tower)**:
- Rearranging the equation to solve for H:
\[ H = 160 \times \sqrt{3} \]
- Now, substituting the approximate value of \(\sqrt{3} \approx 1.732\):
\[ H = 160 \times 1.732 \]
- Calculating the height:
\[ H \approx 277.12 \text{ meters} \]

7. **Final Answer**:
- The height of the tower is approximately \(277.12\) meters.