Two observers are stationed due north of a tower at a distance of 20 m from each other. If the elevations of the tower observed by them are `30^(@) and 45^(@)` respectively, then the height of the tower is

Options

A. 10 m
B. `16 . 32 `
C. `10 (sqrt(3) + 1) m `
D. 30 m

Answer

To find the height of the tower given the angles of elevation from two observers stationed 20 meters apart, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Setup:**
- Let the height of the tower be \( H \).
- Let the distance from the first observer (who sees the angle of elevation \( 45^\circ \)) to the base of the tower be \( x \).
- Therefore, the distance from the second observer (who sees the angle of elevation \( 30^\circ \)) to the base of the tower will be \( x + 20 \) meters.

2. **Using Triangle ABC (Observer 1):**
- In triangle ABC, where angle A is \( 45^\circ \):
\[ \tan(45^\circ) = \frac{H}{x} \]
- Since \( \tan(45^\circ) = 1 \):
\[ 1 = \frac{H}{x} \implies H = x \]

3. **Using Triangle ABD (Observer 2):**
- In triangle ABD, where angle A is \( 30^\circ \):
\[ \tan(30^\circ) = \frac{H}{x + 20} \]
- Since \( \tan(30^\circ) = \frac{1}{\sqrt{3}} \):
\[ \frac{1}{\sqrt{3}} = \frac{H}{x + 20} \]
- Rearranging gives:
\[ H = \frac{x + 20}{\sqrt{3}} \]

4. **Equating the Two Expressions for H:**
- From step 2, we have \( H = x \).
- From step 3, we have \( H = \frac{x + 20}{\sqrt{3}} \).
- Setting these equal:
\[ x = \frac{x + 20}{\sqrt{3}} \]

5. **Cross-Multiplying to Solve for x:**
\[ x \sqrt{3} = x + 20 \]
- Rearranging gives:
\[ x \sqrt{3} - x = 20 \]
- Factoring out \( x \):
\[ x(\sqrt{3} - 1) = 20 \]
- Therefore:
\[ x = \frac{20}{\sqrt{3} - 1} \]

6. **Rationalizing the Denominator:**
- Multiply the numerator and denominator by \( \sqrt{3} + 1 \):
\[ x = \frac{20(\sqrt{3} + 1)}{(\sqrt{3} - 1)(\sqrt{3} + 1)} = \frac{20(\sqrt{3} + 1)}{3 - 1} = \frac{20(\sqrt{3} + 1)}{2} = 10(\sqrt{3} + 1) \]

7. **Finding the Height H:**
- Since \( H = x \):
\[ H = 10(\sqrt{3} + 1) \]

### Final Answer:
The height of the tower is \( 10(\sqrt{3} + 1) \) meters.