If the angle of elevation of the top of a summit is `45^@` and a person climbs at an inclination of `30^@` upto 1 km, where the angle of elevation of top becomes `60^@`, then height of the summit is

Options

A. `1/(sqrt(3)+1)`km
B. `(sqrt(3)-2)/2` km
C. `(sqrt(3)+1)/2` km
D. `3/(sqrt(3)-1)` km

Answer

To find the height of the summit given the angles of elevation and the distance climbed, we can follow these steps:

### Step 1: Understand the Problem
We have a person who observes the summit at two different positions. Initially, the angle of elevation is \(45^\circ\), and after climbing at an inclination of \(30^\circ\) for \(1 \text{ km}\), the angle of elevation becomes \(60^\circ\). We need to find the height of the summit.

### Step 2: Set Up the Diagram
1. Let \(A\) be the position where the angle of elevation is \(45^\circ\).
2. Let \(B\) be the position after climbing \(1 \text{ km}\) at an inclination of \(30^\circ\).
3. Let \(C\) be the summit, and let \(h\) be the height of the summit \(C\) from the ground level.
4. Let \(D\) be the point directly below the summit on the ground.

### Step 3: Analyze the First Position (Point A)
From point \(A\):
- The angle of elevation to the summit \(C\) is \(45^\circ\).
- Using the tangent of the angle, we have:
\[ \tan(45^\circ) = \frac{h}{x} \implies 1 = \frac{h}{x} \implies h = x \]
where \(x\) is the horizontal distance from \(A\) to \(D\).

### Step 4: Analyze the Second Position (Point B)
From point \(B\):
- The angle of elevation to the summit \(C\) is \(60^\circ\).
- The person climbs \(1 \text{ km}\) at an angle of \(30^\circ\). The horizontal distance \(y\) can be calculated using:
\[ y = 1 \cdot \cos(30^\circ) = 1 \cdot \frac{\sqrt{3}}{2} = \frac{\sqrt{3}}{2} \text{ km} \]
- The vertical height gained \(z\) is:
\[ z = 1 \cdot \sin(30^\circ) = 1 \cdot \frac{1}{2} = \frac{1}{2} \text{ km} \]
- The new horizontal distance from point \(B\) to point \(D\) is:
\[ x + y = x + \frac{\sqrt{3}}{2} \]
- Using the tangent of the angle at point \(B\):
\[ \tan(60^\circ) = \frac{h + z}{x + y} \implies \sqrt{3} = \frac{h + \frac{1}{2}}{x + \frac{\sqrt{3}}{2}} \]

### Step 5: Substitute \(h\) and Solve
Substituting \(h = x\) into the equation:
\[ \sqrt{3} = \frac{x + \frac{1}{2}}{x + \frac{\sqrt{3}}{2}} \]
Cross-multiplying gives:
\[ \sqrt{3}(x + \frac{\sqrt{3}}{2}) = x + \frac{1}{2} \]
Expanding and rearranging:
\[ \sqrt{3}x + \frac{3}{2} = x + \frac{1}{2} \]
\[ \sqrt{3}x - x = \frac{1}{2} - \frac{3}{2} \]
\[ (\sqrt{3} - 1)x = -1 \implies x = \frac{-1}{\sqrt{3} - 1} \]

### Step 6: Find Height \(h\)
Now substituting back to find \(h\):
\[ h = x = \frac{-1}{\sqrt{3} - 1} \]
To find \(h\), we can rationalize the denominator:
\[ h = \frac{-1(\sqrt{3} + 1)}{(\sqrt{3} - 1)(\sqrt{3} + 1)} = \frac{-\sqrt{3} - 1}{2} \]

### Final Calculation
Since height cannot be negative, we take the absolute value:
\[ h = \frac{\sqrt{3} + 1}{2} \text{ km} \]

### Conclusion
The height of the summit is:
\[ h = \frac{\sqrt{3} + 1}{2} \text{ km} \]