An observer, `1.5 m` tall, is `20.5 m` away from a tower `22 m` high. Determine the angle of elevation of the top of the tower from the eye of the observer.

Options

A. `60^@`
B. `45^@`
C. `30^@`
D. `90^@`

Explanation

To solve the problem, we need to determine the angle of elevation of the top of the tower from the eye level of the observer. Let's break down the solution step by step.

### Step 1: Understand the scenario
We have:
- Height of the observer (AB) = 1.5 m
- Height of the tower (BC) = 22 m
- Distance from observer to tower (AD) = 20.5 m

### Step 2: Determine the height from the observer's eye level to the top of the tower
To find the height of the tower above the observer's eye level, we subtract the height of the observer from the height of the tower:
\[ DE = BC - AB = 22 \, \text{m} - 1.5 \, \text{m} = 20.5 \, \text{m} \]

### Step 3: Set up the triangle for angle of elevation
Now, we can visualize the situation as a right triangle ADE, where:
- DE is the height from the observer's eye level to the top of the tower (20.5 m)
- AD is the horizontal distance from the observer to the tower (20.5 m)

### Step 4: Use the tangent function to find the angle of elevation
The tangent of the angle of elevation (θ) can be expressed as:
\[ \tan(\theta) = \frac{\text{Opposite}}{\text{Adjacent}} = \frac{DE}{AD} \]
Substituting the values we found:
\[ \tan(\theta) = \frac{20.5 \, \text{m}}{20.5 \, \text{m}} = 1 \]

### Step 5: Determine the angle θ
To find the angle θ, we use the inverse tangent function:
\[ \theta = \tan^{-1}(1) \]
The angle whose tangent is 1 is:
\[ \theta = 45^\circ \]

### Conclusion
The angle of elevation of the top of the tower from the eye of the observer is:
\[ \theta = 45^\circ \]

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