A tower standing on a horizontal plane subtends a certain angle at a point 160 m apart from the foot of the tower .On advancing 100 m towards it , the tower is found to subtend an angle twice as be fore .The height of the tower is

Options

A. 80 m
B. 100 m
C. 160 m
D. 200 m

Answer

To solve the problem, we can use trigonometric principles involving angles of elevation and right triangles. Let's denote:

- \( h \) = height of the tower
- \( A \) = point at the foot of the tower
- \( B \) = initial point where the angle is measured (160 m from A)
- \( C \) = point after advancing 100 m towards the tower (60 m from A)

### Step 1: Define the angles
Let \( \theta \) be the angle subtended by the tower at point B (160 m from the foot of the tower). When the observer moves to point C (60 m from the foot of the tower), the angle subtended becomes \( 2\theta \).

### Step 2: Use the tangent function
From point B, we can express the height of the tower in terms of \( \theta \):
\[ \tan(\theta) = \frac{h}{160} \]
From point C, the height can be expressed as:
\[ \tan(2\theta) = \frac{h}{60} \]

### Step 3: Use the double angle formula
Using the double angle formula for tangent:
\[ \tan(2\theta) = \frac{2\tan(\theta)}{1 - \tan^2(\theta)} \]
Substituting \( \tan(\theta) = \frac{h}{160} \):
\[ \tan(2\theta) = \frac{2 \cdot \frac{h}{160}}{1 - \left(\frac{h}{160}\right)^2} \]

### Step 4: Set up the equation
Now we can set the two expressions for \( \tan(2\theta) \) equal to each other:
\[ \frac{2 \cdot \frac{h}{160}}{1 - \left(\frac{h}{160}\right)^2} = \frac{h}{60} \]

### Step 5: Cross-multiply and simplify
Cross-multiplying gives us:
\[ 2h \cdot 60 = h \cdot 160 \cdot \left(1 - \left(\frac{h}{160}\right)^2\right) \]
This simplifies to:
\[ 120h = 160h - \frac{h^3}{160} \]
Rearranging gives:
\[ \frac{h^3}{160} = 40h \]
Multiplying through by 160:
\[ h^3 = 6400h \]

### Step 6: Factor out h
Factoring out \( h \):
\[ h(h^2 - 6400) = 0 \]
This gives us:
\[ h = 0 \quad \text{or} \quad h^2 = 6400 \]
Thus:
\[ h = 80 \quad \text{(since height cannot be negative)} \]

### Conclusion
The height of the tower is \( \boxed{80 \text{ m}} \).