Two men on either side of a temple 126 m high observe the angle of elevation of the top of the temple to be `30^(@) and 60^(@)` respectively. Find the distance between the two men ?

Answer

To solve the problem step by step, we will use trigonometric ratios to find the distances from each man to the base of the temple and then add these distances to find the total distance between the two men.

### Step 1: Understand the problem
We have a temple that is 126 meters high. There are two men, one observing the top of the temple at an angle of elevation of \(30^\circ\) and the other at an angle of elevation of \(60^\circ\).

### Step 2: Set up the diagram
Let:
- Point A be the top of the temple.
- Point O be the base of the temple.
- Point C be the position of the first man (observing at \(30^\circ\)).
- Point B be the position of the second man (observing at \(60^\circ\)).

### Step 3: Define the distances
Let:
- \(OC = x\) (the distance from the first man to the base of the temple).
- \(OB = y\) (the distance from the second man to the base of the temple).

### Step 4: Use trigonometry for the first man (angle of elevation = \(30^\circ\))
In triangle \(AOC\):
- The height of the temple \(AO = 126\) m.
- The angle of elevation \( \angle AOC = 30^\circ\).

Using the tangent function:
\[ \tan(30^\circ) = \frac{AO}{OC} = \frac{126}{x} \]
We know that \( \tan(30^\circ) = \frac{1}{\sqrt{3}} \):
\[ \frac{1}{\sqrt{3}} = \frac{126}{x} \]
Cross-multiplying gives:
\[ x = 126 \sqrt{3} \quad \text{(1)} \]

### Step 5: Use trigonometry for the second man (angle of elevation = \(60^\circ\))
In triangle \(AOB\):
- The height of the temple \(AO = 126\) m.
- The angle of elevation \( \angle AOB = 60^\circ\).

Using the tangent function:
\[ \tan(60^\circ) = \frac{AO}{OB} = \frac{126}{y} \]
We know that \( \tan(60^\circ) = \sqrt{3} \):
\[ \sqrt{3} = \frac{126}{y} \]
Cross-multiplying gives:
\[ y = \frac{126}{\sqrt{3}} = 126 \cdot \frac{\sqrt{3}}{3} = 42 \sqrt{3} \quad \text{(2)} \]

### Step 6: Calculate the total distance between the two men
The total distance \(d\) between the two men is:
\[ d = x + y \]
Substituting the values from (1) and (2):
\[ d = 126 \sqrt{3} + 42 \sqrt{3} = (126 + 42) \sqrt{3} = 168 \sqrt{3} \text{ meters} \]

### Final Answer
The distance between the two men is \(168 \sqrt{3}\) meters.

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