From the top of a 10 m high building, the angle of elevation of the top of a tower is `60^@` and the angle of depression of its foot is `45^@`. Find the height of the tower

Options

A. 17.42
B. 27.32 m
C. 23.62 m
D. none to these

Answer

To find the height of the tower, we will follow these steps:

### Step 1: Understand the problem
We have a building that is 10 meters high. From the top of this building, we have two angles:
- The angle of elevation to the top of the tower is \(60^\circ\).
- The angle of depression to the foot of the tower is \(45^\circ\).

### Step 2: Set up the diagram
Let:
- Point A be the top of the building.
- Point B be the foot of the building.
- Point C be the top of the tower.
- Point D be the foot of the tower.

The height of the building (AB) is 10 m. We need to find the height of the tower (CD).

### Step 3: Identify the right triangles
1. Triangle ABE (where E is the foot of the tower):
- The angle of depression from A to B is \(45^\circ\).
- The height AB = 10 m.

2. Triangle ACE:
- The angle of elevation from A to C is \(60^\circ\).

### Step 4: Calculate the distance BD (the horizontal distance from the building to the tower)
Using triangle ABE:
- Since the angle of depression is \(45^\circ\), we can use the tangent function:
\[ \tan(45^\circ) = \frac{AB}{BD} \]
\[ 1 = \frac{10}{BD} \implies BD = 10 \text{ m} \]

### Step 5: Calculate the height CE (the height of the tower above the building)
Using triangle ACE:
- Here, we will use the tangent function again:
\[ \tan(60^\circ) = \frac{CE}{AB} \]
\[ \sqrt{3} = \frac{CE}{10} \]
\[ CE = 10 \cdot \sqrt{3} \approx 10 \cdot 1.732 = 17.32 \text{ m} \]

### Step 6: Calculate the total height of the tower (CD)
The total height of the tower (CD) is the sum of the height of the building (AB) and the height of the tower above the building (CE):
\[ CD = AB + CE = 10 + 17.32 = 27.32 \text{ m} \]

### Final Answer
The height of the tower is \(27.32\) meters.

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