An observer finds that the angular elevation of a tower is `theta`. On advancing 3m towards the tower, the elevation is `45^(@)` and on advancing 2m further more towards the tower, the elevation is `90^(@)-theta`. The height of the tower is (assume the height of observer is negligible and observer lies on the same level as the foot of the tower)

Options

A. 2m
B. 4m
C. 6m
D. 8m

Answer

To find the height of the tower based on the given information, we can break down the problem step by step.

### Step 1: Understand the scenario
Let the height of the tower be \( h \). The observer is initially at a distance \( x + 5 \) meters from the tower, where \( x \) is the distance from the observer to the foot of the tower when the angle of elevation is \( \theta \).

### Step 2: Set up the first triangle
When the observer moves 3 meters closer to the tower, the distance to the tower becomes \( x + 2 \) meters. The angle of elevation at this point is \( 45^\circ \).

Using the tangent function:
\[ \tan(45^\circ) = \frac{h}{x + 2} \]
Since \( \tan(45^\circ) = 1 \):
\[ h = x + 2 \quad \text{(Equation 1)} \]

### Step 3: Set up the second triangle
When the observer moves another 2 meters closer, the distance to the tower becomes \( x \) meters, and the angle of elevation is \( 90^\circ - \theta \).

Using the tangent function:
\[ \tan(90^\circ - \theta) = \cot(\theta) = \frac{h}{x} \]
Thus, we have:
\[ h = x \cdot \cot(\theta) \quad \text{(Equation 2)} \]

### Step 4: Relate the two equations
From Equation 1, we have:
\[ x = h - 2 \]
Substituting this into Equation 2 gives:
\[ h = (h - 2) \cdot \cot(\theta) \]

### Step 5: Express cotangent in terms of height
We know that:
\[ \cot(\theta) = \frac{h}{h + 3} \]
Substituting this into the equation gives:
\[ h = (h - 2) \cdot \frac{h}{h + 3} \]

### Step 6: Solve for \( h \)
Cross-multiplying gives:
\[ h(h + 3) = (h - 2)h \]
Expanding both sides:
\[ h^2 + 3h = h^2 - 2h \]
Simplifying:
\[ h^2 + 3h - h^2 + 2h = 0 \]
\[ 5h = 6 \]
Thus:
\[ h^2 = 6 \]

### Step 7: Final height of the tower
The height of the tower is:
\[ h = 6 \text{ meters} \]

### Conclusion
The height of the tower is \( 6 \) meters.

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