The angles of elevation of the top of a tower 72 metre high from the top and bottom of a building are `30^(@) and 60^(@)` respectively. What is the height (in meters) of building ?

Options

A. 42
B. `20sqrt(3)`
C. `24sqrt(3)`
D. 48

Answer

To solve the problem of finding the height of the building given the angles of elevation to the top of a tower, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Problem**:
We have a tower of height 72 meters and two angles of elevation to the top of the tower from the top and bottom of a building. The angle from the bottom of the building is 60 degrees, and the angle from the top of the building is 30 degrees.

2. **Draw a Diagram**:
Draw a vertical line for the tower (AB) and a vertical line for the building (CD). Mark the height of the tower (AB) as 72 meters. Let the height of the building (CD) be 'h' meters.

3. **Identify Points**:
Let:
- A = top of the tower
- B = bottom of the tower
- C = top of the building
- D = bottom of the building

4. **Use Triangle ABC** (from the bottom of the building):
- In triangle ABC, angle ACB = 60 degrees.
- The height of the tower (AB) = 72 meters.
- Using the tangent function:
\[ \tan(60^\circ) = \frac{AB}{BC} \]
where BC is the distance from the bottom of the building to the base of the tower.
- Since \(\tan(60^\circ) = \sqrt{3}\), we have:
\[ \sqrt{3} = \frac{72}{BC} \]
- Rearranging gives:
\[ BC = \frac{72}{\sqrt{3}} = 24\sqrt{3} \text{ meters} \]

5. **Use Triangle ACD** (from the top of the building):
- In triangle ACD, angle ACD = 30 degrees.
- Using the tangent function:
\[ \tan(30^\circ) = \frac{AB - CD}{DC} \]
where DC is the same distance BC.
- Since \(\tan(30^\circ) = \frac{1}{\sqrt{3}}\), we have:
\[ \frac{1}{\sqrt{3}} = \frac{72 - h}{24\sqrt{3}} \]
- Cross-multiplying gives:
\[ 24\sqrt{3} = \sqrt{3}(72 - h) \]
- Simplifying gives:
\[ 24 = 72 - h \]
- Rearranging gives:
\[ h = 72 - 24 = 48 \text{ meters} \]

6. **Conclusion**:
The height of the building (CD) is 48 meters.

### Final Answer:
The height of the building is **48 meters**.