From the top of a building 60 metre high ,the angles of depression of the top and bottom of a tower are observed to be `30^(@)and60^(@)` respectively.The height of the tower in metre is :

Options

A. a.40
B. b.45
C. c.50
D. d.55

Answer

To find the height of the tower, we can follow these steps:

### Step 1: Understand the Problem
We have a building that is 60 meters high. From the top of this building, we observe the angles of depression to the top and bottom of a tower. The angle of depression to the top of the tower is 30 degrees, and to the bottom of the tower is 60 degrees.

### Step 2: Set Up the Diagram
Let's denote:
- Point A as the top of the building (60 meters high).
- Point B as the bottom of the tower.
- Point C as the top of the tower.
- Point D as the base of the building (ground level).

### Step 3: Identify the Angles
From point A (top of the building):
- The angle of depression to point C (top of the tower) is 30 degrees.
- The angle of depression to point B (bottom of the tower) is 60 degrees.

### Step 4: Use Trigonometry for Triangle ABC
In triangle ABC, where:
- AB = 60 meters (height of the building)
- BC = distance from the building to the tower (unknown)
- Angle CAB = 30 degrees

Using the tangent function:
\[ \tan(30^\circ) = \frac{AB}{BC} \]
\[ \tan(30^\circ) = \frac{60}{BC} \]
Since \(\tan(30^\circ) = \frac{1}{\sqrt{3}}\):
\[ \frac{1}{\sqrt{3}} = \frac{60}{BC} \]
Cross-multiplying gives:
\[ BC = 60\sqrt{3} \]

### Step 5: Use Trigonometry for Triangle ABD
In triangle ABD, where:
- AD = height of the tower (unknown)
- BD = distance from the bottom of the tower to the base of the building (same as BC)
- Angle DAB = 60 degrees

Using the tangent function:
\[ \tan(60^\circ) = \frac{AB}{BD} \]
\[ \tan(60^\circ) = \frac{60}{BD} \]
Since \(\tan(60^\circ) = \sqrt{3}\):
\[ \sqrt{3} = \frac{60}{BD} \]
Cross-multiplying gives:
\[ BD = \frac{60}{\sqrt{3}} = 20\sqrt{3} \]

### Step 6: Calculate the Height of the Tower
The height of the tower (DC) can be found by subtracting the height from the top of the tower to the ground (AE) from the height of the building (AB):
\[ DC = AB - AE \]
Where:
- AE = height from the top of the building to the bottom of the tower = 60 - (height of the tower)
- AE = height of the building - height of the tower

Using the values we calculated:
\[ DC = 60 - 20 = 40 \text{ meters} \]

### Final Answer
The height of the tower is **40 meters**.

---