At the foot of the mountain the elevation of its summit is `45^@`, after ascending 100 m towards the mountain up a slope of `30^@` inclination, the elevation is found to be `60^@`. The height of the mountain is

Options

A. `(sqrt3+1)/2`m
B. `(sqrt3-1)/3` m
C. `(sqrt3+1)/(2sqrt3)` m
D. none of these

Answer

To solve the problem, we will break it down step by step.

### Step 1: Understand the Problem
We need to find the height of the mountain given the angles of elevation and the distance traveled along a slope.

### Step 2: Draw a Diagram
Draw a diagram to visualize the situation:
- Let A be the point at the foot of the mountain.
- Let B be the point after ascending 100 m along the slope.
- Let C be the summit of the mountain.
- The angle of elevation from A to C is 45°.
- The angle of elevation from B to C is 60°.
- The slope AB makes an angle of 30° with the horizontal.

### Step 3: Identify the Angles
From the diagram:
- The angle ∠CAB = 45° (angle of elevation from A).
- The angle ∠ABC = 30° (slope inclination).
- The angle ∠BCM = 60° (angle of elevation from B).

### Step 4: Calculate the Remaining Angles
Using the triangle properties:
- In triangle ABC, the angle ∠CBA = 180° - (∠CAB + ∠ABC) = 180° - (45° + 30°) = 105°.
- In triangle ABE, where E is the point vertically below C on the horizontal line through A, we can find the height.

### Step 5: Use Trigonometric Ratios
Using the sine rule in triangle ABE:
- For triangle ABE:
\[ BE = AB \cdot \sin(30°) = 100 \cdot \frac{1}{2} = 50 \text{ m} \]

Using the cosine rule in triangle ABE:
- For triangle ABE:
\[ AE = AB \cdot \cos(30°) = 100 \cdot \frac{\sqrt{3}}{2} = 50\sqrt{3} \text{ m} \]

### Step 6: Calculate Height from B to C
Using the angle of elevation from B to C:
- In triangle BEC:
\[ CE = BE \cdot \tan(60°) = 50 \cdot \sqrt{3} \text{ m} \]

### Step 7: Calculate Total Height of the Mountain
The total height of the mountain (H) is the sum of the height from A to E and from B to C:
- Total height \( H = AE + CE \)
- \( H = 50\sqrt{3} + 50\sqrt{3} = 100\sqrt{3} \text{ m} \)

### Final Answer
The height of the mountain is \( 100\sqrt{3} \text{ m} \).

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