Looking from the top of a 20 m high building, the angle of elevation of the top of a tower is `60^(@)` and the angle of depression of its bottom is `30^(@)`. What is the height of the tower?

Options

A. 50m
B. 60m
C. 70m
D. 80m

Explanation

To solve the problem step by step, we will use trigonometric ratios and the properties of right triangles formed by the angles of elevation and depression.

### Step 1: Understand the Problem
We have a building of height 20 m. From the top of this building, the angle of elevation to the top of a tower is 60 degrees, and the angle of depression to the bottom of the tower is 30 degrees. We need to find the height of the tower.

### Step 2: Define Variables
- Let the height of the tower be \( H \).
- The height of the building is given as 20 m.
- Let the distance from the base of the building to the base of the tower be \( x \).

### Step 3: Analyze the Triangle for the Angle of Elevation
From the top of the building, we can form a right triangle when looking at the top of the tower:
- The opposite side (height of the tower above the building) is \( H - 20 \).
- The adjacent side is \( x \).
- The angle of elevation is 60 degrees.

Using the tangent function:
\[ \tan(60^\circ) = \frac{H - 20}{x} \]
Since \( \tan(60^\circ) = \sqrt{3} \), we have:
\[ \sqrt{3} = \frac{H - 20}{x} \]
This can be rearranged to:
\[ H - 20 = \sqrt{3} x \quad \text{(1)} \]

### Step 4: Analyze the Triangle for the Angle of Depression
Now, we analyze the triangle formed when looking at the bottom of the tower:
- The opposite side (height of the building) is 20 m.
- The adjacent side is \( x \).
- The angle of depression is 30 degrees.

Using the tangent function:
\[ \tan(30^\circ) = \frac{20}{x} \]
Since \( \tan(30^\circ) = \frac{1}{\sqrt{3}} \), we have:
\[ \frac{1}{\sqrt{3}} = \frac{20}{x} \]
This can be rearranged to:
\[ x = 20\sqrt{3} \quad \text{(2)} \]

### Step 5: Substitute \( x \) into Equation (1)
Now we substitute equation (2) into equation (1):
\[ H - 20 = \sqrt{3} (20\sqrt{3}) \]
\[ H - 20 = 60 \]
Adding 20 to both sides:
\[ H = 80 \]

### Step 6: Conclusion
The height of the tower is \( H = 80 \) meters.

### Summary
The height of the tower is 80 meters.