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 `45^(@)`. If the tower is 30 m high, find the height of the building.

Answer

To solve the problem, we will use trigonometric ratios and the properties of right triangles. Let's break it down step by step.

### Step 1: Understand the Problem
We have a tower and a building. The height of the tower (let's denote it as \( h_T \)) is given as 30 m. The angle of elevation from the foot of the tower to the top of the building is \( 30^\circ \), and the angle of elevation from the foot of the building to the top of the tower is \( 45^\circ \). We need to find the height of the building (let's denote it as \( h_B \)).

### Step 2: Set Up the Diagram
1. Let \( A \) be the foot of the tower.
2. Let \( B \) be the top of the tower.
3. Let \( C \) be the foot of the building.
4. Let \( D \) be the top of the building.

We have:
- \( AB = h_T = 30 \) m (height of the tower)
- \( \angle CAB = 30^\circ \) (angle of elevation from \( A \) to \( D \))
- \( \angle BCA = 45^\circ \) (angle of elevation from \( C \) to \( B \))

### Step 3: Use Triangle \( ABC \)
In triangle \( ABC \):
- \( AB \) is the opposite side (height of the tower).
- \( AC \) is the adjacent side (distance from the foot of the tower to the foot of the building).

Using the tangent function:
\[ \tan(45^\circ) = \frac{AB}{AC} \]
Since \( \tan(45^\circ) = 1 \):
\[ 1 = \frac{30}{AC} \implies AC = 30 \text{ m} \]

### Step 4: Use Triangle \( ACD \)
Now, in triangle \( ACD \):
- \( AD \) is the height of the building, which we need to find.
- \( AC \) is still the adjacent side (30 m).

Using the tangent function again:
\[ \tan(30^\circ) = \frac{AD}{AC} \]
Since \( \tan(30^\circ) = \frac{1}{\sqrt{3}} \):
\[ \frac{1}{\sqrt{3}} = \frac{AD}{30} \]
Multiplying both sides by 30:
\[ AD = \frac{30}{\sqrt{3}} = 10\sqrt{3} \text{ m} \]

### Step 5: Find the Height of the Building
The height of the building \( h_B \) is equal to \( AD \):
\[ h_B = 10\sqrt{3} \text{ m} \]

### Final Answer
The height of the building is \( 10\sqrt{3} \) meters.

---