An observer of length 1.5m is standing 30m away from a tower. Angle of elevation from eye is `45^(@)`, then the height of tower is

Options

A. 30m
B. 31.5m
C. 28.5m
D. 29.5m

Answer

To find the height of the tower based on the given information, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Problem**:
- The observer is 1.5 meters tall and stands 30 meters away from the tower.
- The angle of elevation from the observer's eye level to the top of the tower is 45 degrees.

2. **Draw a Diagram**:
- Draw a right triangle where:
- Point A is the top of the tower.
- Point B is the observer's eye level (1.5 meters above the ground).
- Point C is the base of the tower.
- The distance from the observer to the tower (BC) is 30 meters.
- The height of the observer's eye level from the ground (CE) is 1.5 meters.

3. **Identify the Heights**:
- Let the height of the tower be represented as \( h \).
- The height from the observer's eye level to the top of the tower is \( h - 1.5 \) meters.

4. **Use Trigonometry**:
- In triangle ABC, where angle B is 45 degrees:
- We know that:
\[ \tan(\text{angle B}) = \frac{\text{opposite}}{\text{adjacent}} = \frac{h - 1.5}{30} \]
- Since \( \tan(45^\circ) = 1 \):
\[ 1 = \frac{h - 1.5}{30} \]

5. **Solve for \( h \)**:
- Rearranging the equation gives:
\[ h - 1.5 = 30 \]
- Adding 1.5 to both sides:
\[ h = 30 + 1.5 = 31.5 \text{ meters} \]

6. **Conclusion**:
- The height of the tower is \( 31.5 \) meters.