Horizontal distance between two pillars of different heights is 60 m . It was observed that the angular elevation from the top of the shorter pillar is `45^(@)` . If the height of taller pillar is 130 m, the height of the shorter pillar is

Options

A. 45 m
B. 70 m
C. 80 m
D. 60 m

Answer

To solve the problem step by step, we can follow these instructions:

### Step 1: Understand the Problem
We have two pillars with a horizontal distance of 60 m between them. The height of the taller pillar is 130 m, and we need to find the height of the shorter pillar. The angle of elevation from the top of the shorter pillar to the top of the taller pillar is 45°.

### Step 2: Set Up the Diagram
Let's denote:
- Height of the taller pillar (AB) = 130 m
- Height of the shorter pillar (CD) = h m
- Horizontal distance (BC) = 60 m

### Step 3: Use Trigonometry
From the top of the shorter pillar (point C), the angle of elevation to the top of the taller pillar (point A) is 45°. We can use the tangent function to relate the heights and the distance:
\[ \tan(\theta) = \frac{\text{Opposite}}{\text{Adjacent}} \]
Here, \(\theta = 45^\circ\), so:
\[ \tan(45^\circ) = 1 \]
This means:
\[ \frac{AB - CD}{BC} = 1 \]
Substituting the known values:
\[ \frac{130 - h}{60} = 1 \]

### Step 4: Solve for h
Now, we can solve for h:
\[ 130 - h = 60 \]
Adding h to both sides:
\[ 130 = 60 + h \]
Subtracting 60 from both sides:
\[ h = 130 - 60 \]
\[ h = 70 \text{ m} \]

### Step 5: Conclusion
The height of the shorter pillar is 70 m.