A flag-staff 20 m long standing on a wall 10 m high subtends an angle whose tangent is 0.5 at a point on the ground. If `theta` is the angle subtended by the wall at this point, then

Options

A. `tantheta=1`
B. `tantheta=1//3`
C. `tantheta=3`
D. `tantheta=1//2`

Answer

To solve the problem, we need to find the angle \( \theta \) subtended by the wall at a point on the ground, given that a flagstaff of length 20 m is standing on a wall of height 10 m and subtends an angle whose tangent is 0.5 at that point.

### Step-by-Step Solution:

1. **Understanding the Geometry**:
- Let the height of the wall be \( h = 10 \) m.
- The length of the flagstaff is \( l = 20 \) m.
- The total height from the ground to the top of the flagstaff is \( H = h + l = 10 + 20 = 30 \) m.

2. **Finding the Horizontal Distance**:
- The tangent of the angle subtended by the flagstaff at the point on the ground is given as \( \tan(\alpha) = 0.5 \).
- We can use the tangent definition:
\[ \tan(\alpha) = \frac{\text{opposite}}{\text{adjacent}} = \frac{H}{d} \]
where \( d \) is the horizontal distance from the point on the ground to the base of the wall.
- Rearranging gives us:
\[ d = \frac{H}{\tan(\alpha)} = \frac{30}{0.5} = 60 \text{ m} \]

3. **Finding the Angle \( \theta \)**:
- Now, we need to find the angle \( \theta \) subtended by the wall at the same point.
- The tangent of angle \( \theta \) can be expressed as:
\[ \tan(\theta) = \frac{h}{d} = \frac{10}{60} = \frac{1}{6} \]

4. **Calculating \( \theta \)**:
- To find \( \theta \), we can use the arctangent function:
\[ \theta = \tan^{-1}\left(\frac{1}{6}\right) \]

5. **Conclusion**:
- The angle \( \theta \) can be calculated using a calculator or trigonometric tables to find the value of \( \tan^{-1}\left(\frac{1}{6}\right) \).

### Final Answer:
The angle \( \theta \) subtended by the wall at the point on the ground is \( \tan^{-1}\left(\frac{1}{6}\right) \).