A boy, 1.6 m tall, is 20 m away from a tower and observes the angle of elevation of the top of the tower to be `45^(@)` then find the height of tower

Answer

To solve the problem step by step, we can follow these instructions:

### Step 1: Understand the problem
We have a boy who is 1.6 m tall standing 20 m away from a tower. The angle of elevation from the boy's eyes to the top of the tower is 45 degrees. We need to find the height of the tower.

### Step 2: Draw a diagram
Let's represent the situation with a diagram:
- Let AB be the height of the tower.
- Let CD be the height of the boy, which is 1.6 m.
- Let E be the point where the boy is standing.
- The distance from the boy to the tower (point D to point B) is 20 m.

### Step 3: Identify the points and distances
- The height of the boy (CD) = 1.6 m.
- The distance from the boy to the tower (DE) = 20 m.
- The angle of elevation (∠ACE) = 45 degrees.

### Step 4: Use the tangent function
In triangle ACE, we can use the tangent of the angle of elevation:
\[ \tan(45^\circ) = \frac{AE}{CE} \]
Where:
- AE is the height of the tower above the boy's height.
- CE is the horizontal distance from the boy to the tower, which is 20 m.

Since \(\tan(45^\circ) = 1\), we have:
\[ 1 = \frac{AE}{20} \]

### Step 5: Solve for AE
From the equation above, we can find AE:
\[ AE = 20 \times 1 = 20 \text{ m} \]

### Step 6: Calculate the total height of the tower
The total height of the tower (AB) is the height from the ground to the top of the tower, which is the height above the boy (AE) plus the height of the boy (CD):
\[ AB = AE + CD \]
Substituting the values:
\[ AB = 20 \text{ m} + 1.6 \text{ m} = 21.6 \text{ m} \]

### Final Answer
The height of the tower is **21.6 meters**.

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