The angle of elevation of the top of a building from the foot of the tower is `30^@`and the angle of elevation of the top of the tower from the foot of the building is `60^@`. If the tower is `50` m high, find the height of the building.

Options

A. `10/3`
B. `20/3`
C. `50/3`
D. `40/3`

Explanation

To solve the problem, we will use trigonometric ratios and properties of right-angled triangles.

### Step-by-Step Solution:

1. **Understand the Problem:**
We have a tower of height 50 m and we need to find the height of the building. The angle of elevation from the foot of the tower to the top of the building is 30°, and the angle of elevation from the foot of the building to the top of the tower is 60°.

2. **Label the Points:**
- Let point A be the top of the building.
- Let point B be the foot of the building.
- Let point C be the top of the tower.
- Let point D be the foot of the tower.

The height of the tower (CD) = 50 m.

3. **Identify the Triangles:**
We will consider two right-angled triangles:
- Triangle DBC (where D is the foot of the tower, B is the foot of the building, and C is the top of the tower).
- Triangle ABC (where A is the top of the building, B is the foot of the building, and C is the top of the tower).

4. **Find the Distance BC Using Triangle DBC:**
In triangle DBC, we know:
- Angle DBC = 60° (angle of elevation to the top of the tower).
- Height CD = 50 m.

Using the tangent function:
\[ \tan(60°) = \frac{CD}{BC} \]
\[ \sqrt{3} = \frac{50}{BC} \]
Rearranging gives:
\[ BC = \frac{50}{\sqrt{3}} \text{ m} \]

5. **Find the Height AB Using Triangle ABC:**
In triangle ABC, we know:
- Angle ABC = 30° (angle of elevation to the top of the building).
- We need to find the height AB (height of the building).

Using the tangent function:
\[ \tan(30°) = \frac{AB}{BC} \]
\[ \frac{1}{\sqrt{3}} = \frac{AB}{\frac{50}{\sqrt{3}}} \]
Rearranging gives:
\[ AB = \frac{50}{\sqrt{3}} \cdot \frac{1}{\sqrt{3}} = \frac{50}{3} \text{ m} \]

6. **Conclusion:**
The height of the building (AB) is \(\frac{50}{3}\) meters.

### Final Answer:
The height of the building is \(\frac{50}{3}\) meters.

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