At the foot of a mountain, the elevation of its summit is `45^@`. After ascending 2 km towards the mountain upon an incline of `30^@`, the elevation changes to `60^@`. The height of the mountain is

Options

A. `(sqrt3 -1)km`
B. `(sqrt3+1)km`
C. `(sqrt3-2) km`
D. `(sqrt3 - 2)km`

Answer

To solve the problem step by step, we will use trigonometric concepts related to heights and distances.

### Step 1: Understand the Problem
We have a mountain with an elevation angle of 45 degrees from the foot of the mountain. After ascending 2 km at an incline of 30 degrees, the elevation angle changes to 60 degrees. We need to find the height of the mountain.

### Step 2: Set Up the Diagram
Let:
- Point A be the foot of the mountain.
- Point B be the summit of the mountain.
- Point C be the point after ascending 2 km at a 30-degree incline.
- Point D be the point directly below B on the horizontal line from A.

### Step 3: Determine the Height from Point C
From point C, the elevation angle to the summit (point B) is 60 degrees. The height of the mountain (AB) can be expressed in terms of the distance from point C to point D (CD) and the angle of elevation (60 degrees).

Using the tangent function:
\[ \tan(60^\circ) = \frac{AB}{CD} \]
Since \(\tan(60^\circ) = \sqrt{3}\), we have:
\[ \sqrt{3} = \frac{AB}{CD} \quad \text{(1)} \]

### Step 4: Calculate CD
From point A to point C, we ascend 2 km at a 30-degree incline. We can find the vertical height (CD) using the sine function:
\[ CD = 2 \cdot \sin(30^\circ) \]
Since \(\sin(30^\circ) = \frac{1}{2}\):
\[ CD = 2 \cdot \frac{1}{2} = 1 \text{ km} \quad \text{(2)} \]

### Step 5: Substitute CD into Equation (1)
Substituting \(CD = 1\) km into equation (1):
\[ \sqrt{3} = \frac{AB}{1} \]
Thus, we find:
\[ AB = \sqrt{3} \text{ km} \quad \text{(3)} \]

### Step 6: Calculate the Height of the Mountain
Now we need to find the total height of the mountain, which is the height from point A to point B. Since we already calculated the height from point C to point B (AB) and the height from point A to point C (CD):
\[ \text{Total height} = AB + CD \]
Substituting the values from (2) and (3):
\[ \text{Total height} = \sqrt{3} + 1 \text{ km} \]

### Final Answer
The height of the mountain is:
\[ \sqrt{3} + 1 \text{ km} \]