There are two vertical posts ,one on each side of a road ,just opposite to each other .One post is 108 metre high .From the top of this post , the angle of depression of the top and foot of the other post are `30^(@)and60^(@)` respectively .The height of the other post (in metre ) is

Options

A. 36
B. 72
C. 108
D. 110

Answer

To solve the problem step by step, we will use trigonometric concepts related to angles of depression and right triangles.

### Step 1: Understand the Problem
We have two vertical posts. The height of the first post (let's call it Post A) is 108 meters. From the top of Post A, the angles of depression to the top and bottom of the second post (Post B) are 30 degrees and 60 degrees, respectively. We need to find the height of Post B.

### Step 2: Draw a Diagram
Draw a diagram with two vertical posts. Label the height of Post A as AB = 108 meters. Let the height of Post B be h meters. The horizontal distance between the two posts can be labeled as BC.

### Step 3: Identify the Right Triangles
From the top of Post A (point A), draw lines to the top (point D) and bottom (point C) of Post B. This creates two right triangles:
1. Triangle ABC (for the angle of depression to the foot of Post B)
2. Triangle AED (for the angle of depression to the top of Post B)

### Step 4: Use the Angle of Depression to Find Distances
1. **For Triangle ABC** (angle of depression = 60 degrees):
- The angle at point B (angle ABC) is also 60 degrees (alternate interior angles).
- Using the tangent function:
\[ \tan(60^\circ) = \frac{AB}{BC} \]
\[ \sqrt{3} = \frac{108}{BC} \]
Rearranging gives:
\[ BC = \frac{108}{\sqrt{3}} = 36\sqrt{3} \text{ meters} \]

2. **For Triangle AED** (angle of depression = 30 degrees):
- The angle at point E (angle AED) is also 30 degrees.
- Using the tangent function:
\[ \tan(30^\circ) = \frac{AE}{ED} \]
\[ \frac{1}{\sqrt{3}} = \frac{108 - h}{BC} \]
Substituting the value of BC:
\[ \frac{1}{\sqrt{3}} = \frac{108 - h}{36\sqrt{3}} \]
Cross-multiplying gives:
\[ 36\sqrt{3} = \sqrt{3}(108 - h) \]
Simplifying:
\[ 36 = 108 - h \]
Rearranging gives:
\[ h = 108 - 36 = 72 \text{ meters} \]

### Conclusion
The height of the second post (Post B) is **72 meters**.