The persons are standing on the opposite sides of a tower. They observe the angles of elevation of the top of the tower to be `30^(@) and 38^(@)` respectively. Find the distance between them, if the height of the tower is 50 m

Answer

To solve the problem step by step, we will use the concepts of trigonometry, specifically the tangent function, which relates the angles of elevation to the height and distance from the tower.

### Step-by-Step Solution:

1. **Identify the Given Information:**
- Height of the tower (PQ) = 50 m
- Angle of elevation from point A (QA) = 30°
- Angle of elevation from point B (BQ) = 38°

2. **Set Up the Triangles:**
- Let \( QA = x \) (distance from person A to the tower).
- Let \( BQ = y \) (distance from person B to the tower).

3. **Using Triangle PQA (for angle 30°):**
- In triangle PQA, we can use the tangent function:
\[ \tan(30°) = \frac{\text{Opposite}}{\text{Adjacent}} = \frac{PQ}{QA} = \frac{50}{x} \]
- We know that \( \tan(30°) = \frac{1}{\sqrt{3}} \).
- Therefore, we can set up the equation:
\[ \frac{1}{\sqrt{3}} = \frac{50}{x} \]
- Cross-multiplying gives:
\[ x = 50\sqrt{3} \]
- Calculating \( x \):
\[ x \approx 50 \times 1.732 = 86.6 \text{ m} \]

4. **Using Triangle PQB (for angle 38°):**
- In triangle PQB, we apply the tangent function again:
\[ \tan(38°) = \frac{PQ}{BQ} = \frac{50}{y} \]
- We know that \( \tan(38°) \approx 0.7813 \).
- Thus, we set up the equation:
\[ 0.7813 = \frac{50}{y} \]
- Cross-multiplying gives:
\[ y = \frac{50}{0.7813} \]
- Calculating \( y \):
\[ y \approx 64 \text{ m} \]

5. **Finding the Total Distance Between the Two Persons:**
- The total distance \( AB \) between the two persons is the sum of \( QA \) and \( BQ \):
\[ AB = QA + BQ = x + y = 86.6 + 64 \]
- Therefore:
\[ AB \approx 150.6 \text{ m} \]

### Final Answer:
The distance between the two persons is approximately **150.6 meters**.