A flag staff of 5m high stands on a building of 25m high. At an observer at a height of 30 m. The flag staff and the building subtend equal angles . The distance of the observer from the top of the flag staff is

Options

A. `(5sqrt3)/2`
B. `5sqrt3/2`
C. `5sqrt2/3`
D. none of these

Answer

To solve the problem, we need to find the distance of the observer from the top of the flagstaff, given that the flagstaff and the building subtend equal angles at the observer.

### Step-by-Step Solution:

1. **Identify the heights**:
- Height of the flagstaff = 5 m
- Height of the building = 25 m
- Height of the observer = 30 m

2. **Determine the total height of the building with the flagstaff**:
- Total height from the ground to the top of the flagstaff = Height of the building + Height of the flagstaff = 25 m + 5 m = 30 m.

3. **Set up the angles**:
- Let the distance from the observer to the base of the building be \( x \).
- The angle subtended by the building at the observer is \( \theta \).
- The angle subtended by the flagstaff at the observer is also \( \theta \).

4. **Use the tangent function**:
- For the building:
\[ \tan(\theta) = \frac{\text{Height of the building}}{\text{Distance from observer to building}} = \frac{25}{x} \]
- For the flagstaff:
\[ \tan(\theta) = \frac{\text{Height of the flagstaff}}{\text{Distance from observer to flagstaff}} = \frac{5}{d} \]
- Here, \( d \) is the distance from the observer to the top of the flagstaff.

5. **Relate the distances**:
- Since the observer is at a height of 30 m, the distance from the observer to the top of the flagstaff can be expressed as:
\[ d = \sqrt{x^2 + (30 - 5)^2} = \sqrt{x^2 + 25^2} = \sqrt{x^2 + 625} \]

6. **Setting the equations equal**:
- Since both angles are equal, we can equate the two expressions for \( \tan(\theta) \):
\[ \frac{25}{x} = \frac{5}{\sqrt{x^2 + 625}} \]

7. **Cross-multiply to solve for \( x \)**:
- Cross-multiplying gives:
\[ 25 \cdot \sqrt{x^2 + 625} = 5x \]
- Simplifying leads to:
\[ 5 \cdot \sqrt{x^2 + 625} = x \]
- Squaring both sides:
\[ 25(x^2 + 625) = x^2 \]
- Rearranging gives:
\[ 24x^2 = 15625 \]
- Solving for \( x^2 \):
\[ x^2 = \frac{15625}{24} \approx 651.04167 \]

8. **Finding \( d \)**:
- Now substitute \( x \) back to find \( d \):
\[ d = \sqrt{x^2 + 625} = \sqrt{651.04167 + 625} = \sqrt{1276.04167} \approx 35.7 \text{ m} \]

### Final Answer:
The distance of the observer from the top of the flagstaff is approximately **35.7 m**.