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 step by step, we will use trigonometric ratios and the properties of right triangles.

### Step 1: Understand the Problem
We have a tower and a building. The height of the tower (CD) is given as 30 m. We need to find the height of the building (AB). 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 45°, respectively.

### Step 2: Draw the Diagram
Let's label the points:
- Let A be the top of the building.
- Let B be the foot of the building.
- Let C be the top of the tower.
- Let D be the foot of the tower.

The height of the tower (CD) = 30 m.
The angle of elevation from D (foot of the tower) to A (top of the building) = 30°.
The angle of elevation from B (foot of the building) to C (top of the tower) = 45°.

### Step 3: Analyze Triangle BCD
In triangle BCD, we can use the tangent function:
\[ \tan(45°) = \frac{\text{Opposite}}{\text{Adjacent}} = \frac{CD}{BD} \]
Here, CD = 30 m and tan(45°) = 1. Therefore:
\[ 1 = \frac{30}{BD} \implies BD = 30 \text{ m} \]

### Step 4: Analyze Triangle ABD
Now, we will analyze triangle ABD. We can use the tangent function again:
\[ \tan(30°) = \frac{AB}{BD} \]
We already found that BD = 30 m. Since tan(30°) = \( \frac{1}{\sqrt{3}} \), we have:
\[ \frac{1}{\sqrt{3}} = \frac{AB}{30} \]

### Step 5: Solve for AB
Now, we can rearrange the equation to find AB:
\[ AB = 30 \cdot \frac{1}{\sqrt{3}} = \frac{30}{\sqrt{3}} \]
To rationalize the denominator:
\[ AB = \frac{30 \cdot \sqrt{3}}{3} = 10\sqrt{3} \]

### Step 6: Calculate the Numerical Value
Using the approximate value of \( \sqrt{3} \approx 1.732 \):
\[ AB \approx 10 \cdot 1.732 = 17.32 \text{ m} \]

### Final Answer
The height of the building (AB) is approximately **17.32 meters**.

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