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
`60^(@)` . Find the height of the tower in each case

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 60 degrees. We need to find the height of the tower.

### Step 2: Draw a Diagram
Draw a diagram to visualize the situation:
- Let the tower be represented as point A (top) and point B (bottom).
- Let the boy's height be represented by point D (his eyes) and point E (the ground level).
- The distance from the boy to the tower is represented as line segment DB, which is 20 m.
- The height of the boy is DE = 1.6 m.

### Step 3: Identify the Triangle
In triangle ABD:
- AB is the height of the tower.
- DB is the distance from the boy to the tower (20 m).
- Angle ADB is the angle of elevation (60 degrees).

### Step 4: Use the Tangent Function
We can use the tangent function to find the height of the tower above the boy's eye level:
\[ \tan(\text{angle}) = \frac{\text{opposite}}{\text{adjacent}} \]
Here, the opposite side is AB (the height of the tower) and the adjacent side is DB (20 m).

### Step 5: Set Up the Equation
Using the angle of elevation:
\[ \tan(60^\circ) = \frac{AB}{DB} \]
We know that \(\tan(60^\circ) = \sqrt{3}\), so we can write:
\[ \sqrt{3} = \frac{AB}{20} \]

### Step 6: Solve for AB
Now, we can solve for AB:
\[ AB = 20 \times \sqrt{3} \]
Calculating this gives:
\[ AB \approx 20 \times 1.732 = 34.64 \text{ m} \]

### Step 7: Find the Total Height of the Tower
The total height of the tower (AC) is the height above the boy's eye level (AB) plus the height of the boy (DE):
\[ AC = AB + DE \]
Substituting the values:
\[ AC = 34.64 + 1.6 = 36.24 \text{ m} \]

### Final Answer
The height of the tower is approximately **36.24 meters**.

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