From the top of a building AB, 60 metres hight, the angles of depression of the top and bottom of a vertical lamp post CD are observed to be `30^(@)` and `60^(@)`, respectively. Find
(i) the horizontal distance between AB and CD.
(ii) the height of the lamp post.

Answer

To solve the problem step by step, we will use the concepts of angles of depression and trigonometry.

### Given:
- Height of building AB = 60 meters
- Angle of depression to the top of the lamp post (CD) = 30 degrees
- Angle of depression to the bottom of the lamp post (CD) = 60 degrees

### Step 1: Understand the Geometry
From the top of the building (point A), we can visualize two right triangles:
1. Triangle ABD for the bottom of the lamp post (D).
2. Triangle ABE for the top of the lamp post (C).

Let:
- \( x \) = horizontal distance between the building (AB) and the lamp post (CD).
- \( h \) = height of the lamp post (CD).

### Step 2: Calculate the Horizontal Distance (x)
Using triangle ABD:
- The angle of depression to point D is 60 degrees.
- Therefore, angle ADB = 60 degrees (alternate interior angles).

Using the tangent function:
\[ \tan(60^\circ) = \frac{AB}{BD} = \frac{60}{x} \]
From trigonometric values, we know:
\[ \tan(60^\circ) = \sqrt{3} \]
Thus, we can write:
\[ \sqrt{3} = \frac{60}{x} \]
Rearranging gives:
\[ x = \frac{60}{\sqrt{3}} = 20\sqrt{3} \]
Calculating \( x \):
\[ x \approx 20 \times 1.732 = 34.64 \text{ meters} \]

### Step 3: Calculate the Height of the Lamp Post (h)
Using triangle ABE:
- The angle of depression to point C is 30 degrees.
- Therefore, angle AEB = 30 degrees.

Using the tangent function:
\[ \tan(30^\circ) = \frac{AB}{BE} = \frac{60}{x} \]
From trigonometric values, we know:
\[ \tan(30^\circ) = \frac{1}{\sqrt{3}} \]
Thus, we can write:
\[ \frac{1}{\sqrt{3}} = \frac{60}{x} \]
Rearranging gives:
\[ x = 60\sqrt{3} \]
Now, we know that the height of the lamp post (CD) can be calculated as:
\[ CD = DE - CE \]
Where \( DE = AB = 60 \) meters and \( CE = 20 \) meters (from triangle ABE).

So:
\[ h = DE - CE = 60 - 20 = 40 \text{ meters} \]

### Final Answers:
(i) The horizontal distance between AB and CD is \( 34.64 \) meters.
(ii) The height of the lamp post is \( 40 \) meters.