Two men on either sideof a 75 m high building and in line with base of buildig observe the angle of elevation of the top of the building as `30^(@)` and `60^(@)`. Find the distance between the two men. (Use `sqrt(3)=1.73`)

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 building.

### Step 1: Understand the problem
We have a building of height 75 m. Two men are standing on either side of the building, observing the top of the building at angles of elevation of 30° and 60°. We need to find the distance between the two men.

### Step 2: Set up the triangles
Let:
- Point A be the top of the building.
- Point B be the base of the building.
- Point C be the position of the first man (angle of elevation 30°).
- Point D be the position of the second man (angle of elevation 60°).

### Step 3: Use trigonometric ratios for the first man (angle 30°)
In triangle ABC:
- Height of the building (AB) = 75 m
- Angle of elevation (∠ACB) = 30°

Using the tangent function:
\[ \tan(30°) = \frac{AB}{BC} = \frac{75}{x_1} \]
Where \(x_1\) is the distance from the first man to the base of the building.

From trigonometric values, we know:
\[ \tan(30°) = \frac{1}{\sqrt{3}} \]

So, we can write:
\[ \frac{1}{\sqrt{3}} = \frac{75}{x_1} \]

### Step 4: Solve for \(x_1\)
Rearranging the equation gives:
\[ x_1 = 75 \cdot \sqrt{3} \]

### Step 5: Use trigonometric ratios for the second man (angle 60°)
In triangle ACD:
- Height of the building (AC) = 75 m
- Angle of elevation (∠ADC) = 60°

Using the tangent function:
\[ \tan(60°) = \frac{AC}{DC} = \frac{75}{x_2} \]
Where \(x_2\) is the distance from the second man to the base of the building.

From trigonometric values, we know:
\[ \tan(60°) = \sqrt{3} \]

So, we can write:
\[ \sqrt{3} = \frac{75}{x_2} \]

### Step 6: Solve for \(x_2\)
Rearranging the equation gives:
\[ x_2 = \frac{75}{\sqrt{3}} \]

### Step 7: Find the total distance between the two men
The total distance between the two men is:
\[ \text{Distance} = x_1 + x_2 = 75\sqrt{3} + \frac{75}{\sqrt{3}} \]

### Step 8: Simplify the expression
To simplify, we can factor out 75:
\[ \text{Distance} = 75 \left( \sqrt{3} + \frac{1}{\sqrt{3}} \right) \]

Finding a common denominator:
\[ \sqrt{3} + \frac{1}{\sqrt{3}} = \frac{3 + 1}{\sqrt{3}} = \frac{4}{\sqrt{3}} \]

So, the distance becomes:
\[ \text{Distance} = 75 \cdot \frac{4}{\sqrt{3}} = \frac{300}{\sqrt{3}} \]

### Step 9: Rationalize the denominator
\[ \text{Distance} = 100\sqrt{3} \]

### Step 10: Substitute the value of \(\sqrt{3}\)
Using \(\sqrt{3} \approx 1.73\):
\[ \text{Distance} = 100 \cdot 1.73 = 173.2 \text{ m} \]

### Final Answer
The distance between the two men is approximately **173.2 meters**.

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