The angle of elevation of the top of a tower from the bottom of a building is twice that from its top. What is the height of the building, if the height of the tower is 75 m and the angle of elevation of the top of the tower from the bottom of the building is `60^@`?

Options

A. 25 m
B. 37.5 m
C. 50 m
D. 60 m

Answer

To solve the problem, we need to find the height of the building (h) given the height of the tower (75 m) and the angles of elevation.

### Step 1: Understand the problem
We have a tower (PO) of height 75 m and a building (AB) of height h. The angle of elevation from the bottom of the building to the top of the tower is 60 degrees. The angle of elevation from the top of the building to the top of the tower is half of that, which is 30 degrees.

### Step 2: Set up the triangles
1. **Triangle BQP** (where B is the bottom of the building, Q is the top of the building, and P is the top of the tower):
- Here, we will use the angle of elevation of 30 degrees.
- The height PQ = 75 - h (since PQ is the height of the tower minus the height of the building).
- Let BQ be the horizontal distance from the base of the building to the base of the tower.

2. **Triangle AOP** (where A is the bottom of the building, O is the top of the tower, and P is the top of the tower):
- Here, we will use the angle of elevation of 60 degrees.
- The height OP = 75 m (the height of the tower).
- AO is the horizontal distance from the bottom of the building to the top of the tower.

### Step 3: Use trigonometric ratios
1. For triangle BQP:
\[ \tan(30^\circ) = \frac{PQ}{BQ} = \frac{75 - h}{BQ} \]
We know that \(\tan(30^\circ) = \frac{1}{\sqrt{3}}\), so:
\[ \frac{1}{\sqrt{3}} = \frac{75 - h}{BQ} \implies BQ = (75 - h) \sqrt{3} \]

2. For triangle AOP:
\[ \tan(60^\circ) = \frac{OP}{AO} = \frac{75}{AO} \]
We know that \(\tan(60^\circ) = \sqrt{3}\), so:
\[ \sqrt{3} = \frac{75}{AO} \implies AO = \frac{75}{\sqrt{3}} = 25\sqrt{3} \]

### Step 4: Set the two expressions for BQ equal
Since both BQ and AO represent the same horizontal distance:
\[ (75 - h) \sqrt{3} = 25\sqrt{3} \]

### Step 5: Solve for h
1. Divide both sides by \(\sqrt{3}\):
\[ 75 - h = 25 \]
2. Rearranging gives:
\[ h = 75 - 25 = 50 \]

### Conclusion
The height of the building is **50 m**.