The angle of elevation of the top of a tower of height H from the foot of another tower in the same plane is `60^(@)` and the angle of elevation of the top of the second tower from the foot of the first tower is `30^(@).` If is the height of the other tower is h , then which one of the following is correct?

Options

A. A. `H = 2h `
B. B. ` H = sqrt3 h `
C. C. `H = 3h `
D. D. None of these

Answer

To solve the problem, we need to establish the relationship between the heights of the two towers based on the given angles of elevation. Let's break it down step by step.

### Step 1: Understand the problem and draw a diagram
We have two towers:
- Tower 1 with height \( H \)
- Tower 2 with height \( h \)

From the foot of Tower 2, the angle of elevation to the top of Tower 1 is \( 60^\circ \). From the foot of Tower 1, the angle of elevation to the top of Tower 2 is \( 30^\circ \).

### Step 2: Set up the triangles
Let’s denote:
- The distance from the foot of Tower 1 to the foot of Tower 2 as \( x \).

Using the tangent function for the angles of elevation, we can set up two equations based on the right triangles formed.

### Step 3: Apply the tangent function for Tower 2
From the foot of Tower 2:
\[ \tan(60^\circ) = \frac{H}{x} \]
Since \( \tan(60^\circ) = \sqrt{3} \), we have:
\[ \sqrt{3} = \frac{H}{x} \implies H = x \sqrt{3} \quad \text{(1)} \]

### Step 4: Apply the tangent function for Tower 1
From the foot of Tower 1:
\[ \tan(30^\circ) = \frac{h}{x} \]
Since \( \tan(30^\circ) = \frac{1}{\sqrt{3}} \), we have:
\[ \frac{1}{\sqrt{3}} = \frac{h}{x} \implies h = \frac{x}{\sqrt{3}} \quad \text{(2)} \]

### Step 5: Substitute equation (2) into equation (1)
From equation (1), we know \( H = x \sqrt{3} \). We can express \( x \) in terms of \( h \) using equation (2):
\[ x = h \sqrt{3} \]
Now substitute \( x \) back into equation (1):
\[ H = (h \sqrt{3}) \sqrt{3} = 3h \]

### Conclusion
Thus, the relationship between the heights of the towers is:
\[ H = 3h \]

### Final Answer
The correct relationship is \( H = 3h \).

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