The angle of elevation of the top of a building from the foot of the tower is `30^o`and the angle of elevation of the top of the tower from the foot of the building is `60^o`. 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 step by step, we will use trigonometric ratios in right triangles.

### Step 1: Understand the Problem
We have a tower of height 50 m and we need to find the height of the building (let's denote it as H). The angles of elevation from the foot of the tower to the top of the building and from the foot of the building to the top of the tower are given as 30° and 60° respectively.

### Step 2: Set Up the Diagram
Let's denote:
- Point A: Foot of the building
- Point B: Top of the building
- Point C: Foot of the tower
- Point D: Top of the tower

We have:
- Height of the tower (CD) = 50 m
- Height of the building (AB) = H
- Angle of elevation from C to B = 30°
- Angle of elevation from A to D = 60°

### Step 3: Use Triangle ACD
In triangle ACD, we can use the tangent of the angle of elevation (60°) to find the distance from the building to the tower (AC).

Using the tangent ratio:
\[ \tan(60°) = \frac{CD}{AC} \]
Substituting the known values:
\[ \sqrt{3} = \frac{50}{AC} \]
From this, we can solve for AC:
\[ AC = \frac{50}{\sqrt{3}} \text{ m} \]

### Step 4: Use Triangle BCD
Now, in triangle BCD, we can use the tangent of the angle of elevation (30°) to find the height of the building (H).

Using the tangent ratio:
\[ \tan(30°) = \frac{AB}{BC} \]
Substituting the known values:
\[ \frac{1}{\sqrt{3}} = \frac{H}{BC} \]
We need to express BC in terms of AC. Since BC = AC (the horizontal distance is the same), we have:
\[ BC = \frac{50}{\sqrt{3}} \text{ m} \]
Now substituting BC into the equation:
\[ \frac{1}{\sqrt{3}} = \frac{H}{\frac{50}{\sqrt{3}}} \]
Cross-multiplying gives:
\[ H = \frac{50}{\sqrt{3}} \cdot \frac{1}{\sqrt{3}} = \frac{50}{3} \text{ m} \]

### Step 5: Final Result
Calculating the value:
\[ H \approx 16.67 \text{ m} \]

Thus, the height of the building is approximately **16.67 meters**.