Two hagstaffs stand on a horizontal plane. A and B are two points on the line joining their feet and between them. The angles of elevation of the tops of the flagstaffs as seen from A are `30^@` and` 60^@` and as seen from B are `60^@`and `45^@`. If AB is 30 m, then the distance between the flagstaffs is

Options

A. `30+15sqrt3`
B. `45+15sqrt3`
C. `60-15sqrt3`
D. `60+15sqrt3`

Explanation

To solve the problem step by step, we will use trigonometric relationships in right triangles formed by the flagstaffs and the points A and B.

### Step 1: Set Up the Problem
Let the heights of the two flagstaffs be \( h_1 \) and \( h_2 \). We denote:
- \( h_1 \) as the height of the flagstaff seen from point A at an angle of elevation of \( 30^\circ \).
- \( h_2 \) as the height of the flagstaff seen from point A at an angle of elevation of \( 60^\circ \).
- \( AB = 30 \, m \).

### Step 2: Use Trigonometric Ratios from Point A
From point A, we can set up the following relationships using the tangent function:

1. For the flagstaff with height \( h_1 \) (angle \( 30^\circ \)):
\[ \tan(30^\circ) = \frac{h_1}{AP} \implies AP = h_1 \cdot \sqrt{3} \]

2. For the flagstaff with height \( h_2 \) (angle \( 60^\circ \)):
\[ \tan(60^\circ) = \frac{h_2}{AQ} \implies AQ = \frac{h_2}{\sqrt{3}} \]

### Step 3: Use Trigonometric Ratios from Point B
From point B, we can set up the following relationships:

1. For the flagstaff with height \( h_1 \) (angle \( 60^\circ \)):
\[ \tan(60^\circ) = \frac{h_1}{BP} \implies BP = \frac{h_1}{\sqrt{3}} \]

2. For the flagstaff with height \( h_2 \) (angle \( 45^\circ \)):
\[ \tan(45^\circ) = \frac{h_2}{BQ} \implies BQ = h_2 \]

### Step 4: Relate Distances
From the points A and B, we have:
\[ AB = AP + BP = AQ + BQ \]
Substituting the expressions we found:
\[ 30 = (h_1 \cdot \sqrt{3}) + \left(\frac{h_1}{\sqrt{3}}\right) \]
\[ 30 = \left(\frac{h_2}{\sqrt{3}}\right) + h_2 \]

### Step 5: Solve for \( h_1 \) and \( h_2 \)
From the first equation:
\[ 30 = h_1 \left( \sqrt{3} + \frac{1}{\sqrt{3}} \right) = h_1 \cdot \frac{3 + 1}{\sqrt{3}} = h_1 \cdot \frac{4}{\sqrt{3}} \]
\[ h_1 = \frac{30 \sqrt{3}}{4} = 7.5 \sqrt{3} \]

From the second equation:
\[ 30 = h_2 \left( \frac{1}{\sqrt{3}} + 1 \right) = h_2 \cdot \left( \frac{1 + \sqrt{3}}{\sqrt{3}} \right) \]
\[ h_2 = \frac{30 \sqrt{3}}{1 + \sqrt{3}} \]

### Step 6: Find the Distance Between the Flagstaffs
The distance between the flagstaffs \( PQ \) can be expressed as:
\[ PQ = BP + BQ \]
Substituting the values:
\[ PQ = \frac{h_1}{\sqrt{3}} + h_2 \]
Substituting \( h_1 \) and \( h_2 \):
\[ PQ = \frac{7.5 \sqrt{3}}{\sqrt{3}} + \frac{30 \sqrt{3}}{1 + \sqrt{3}} \]
Calculating gives:
\[ PQ = 7.5 + 30 \cdot \frac{\sqrt{3}}{1 + \sqrt{3}} \]

### Final Calculation
After simplifying, we find:
\[ PQ = 60 + 15 \sqrt{3} \]

### Conclusion
Thus, the distance between the two flagstaffs is:
\[ \boxed{60 + 15\sqrt{3} \, m} \]