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. `(5sqrt(3))/2`
B. `5sqrt(3/2)`
C. `5//2`
D. None of these

Answer

To solve the problem step by step, let's break it down:

### Given:
- Height of the flagstaff (PR) = 5 m
- Height of the building (PQ) = 25 m
- Height of the observer (R) = 30 m
- The angles subtended by the flagstaff and the building at the observer's position are equal.

### Objective:
Find the distance (RA) from the observer to the top of the flagstaff.

### Step 1: Draw the Diagram
Draw a vertical line representing the building (PQ) of height 25m, on top of which is the flagstaff (PR) of height 5m. The total height from the ground to the top of the flagstaff is 30m (PQ + PR = 25m + 5m). Mark the observer's position (R) at a height of 30m.

### Step 2: Define the Angles
Let the angle subtended by the flagstaff at the observer be θ. Since both the flagstaff and the building subtend equal angles, the angle subtended by the building is also θ.

### Step 3: Set Up the Tangent Ratios
For the flagstaff:
- In triangle PRA, the height is 5m and the base is RA (which we will denote as x).
- Therefore, we have:
\[ \tan(\theta) = \frac{5}{x} \]

For the building:
- In triangle QRA, the height is 30m (25m + 5m) and the base is RA (which is also x).
- Therefore, we have:
\[ \tan(2\theta) = \frac{30}{x} \]

### Step 4: Use the Double Angle Formula
Using the double angle formula for tangent:
\[ \tan(2\theta) = \frac{2\tan(\theta)}{1 - \tan^2(\theta)} \]
Substituting \(\tan(\theta) = \frac{5}{x}\):
\[ \tan(2\theta) = \frac{2 \cdot \frac{5}{x}}{1 - \left(\frac{5}{x}\right)^2} = \frac{\frac{10}{x}}{1 - \frac{25}{x^2}} = \frac{10}{x} \cdot \frac{x^2}{x^2 - 25} = \frac{10x}{x^2 - 25} \]

### Step 5: Set the Two Expressions for \(\tan(2\theta)\) Equal
Now we can set the two expressions for \(\tan(2\theta)\) equal to each other:
\[ \frac{10x}{x^2 - 25} = \frac{30}{x} \]

### Step 6: Cross Multiply and Simplify
Cross multiplying gives:
\[ 10x^2 = 30(x^2 - 25) \]
Expanding the right side:
\[ 10x^2 = 30x^2 - 750 \]
Rearranging gives:
\[ 0 = 20x^2 - 750 \]
\[ 20x^2 = 750 \]
\[ x^2 = \frac{750}{20} = \frac{75}{2} \]

### Step 7: Solve for x
Taking the square root:
\[ x = \sqrt{\frac{75}{2}} = \frac{5\sqrt{3}}{\sqrt{2}} = \frac{5\sqrt{3}}{2} \]

### Final Answer
The distance of the observer from the top of the flagstaff (RA) is:
\[ \frac{5\sqrt{3}}{2} \text{ meters} \]