Two poles of equal heights are standing opposite to each other on either side of the road which is 80 m wide. From a point P between them on the road, the angle of elevation of the top of one pole is `60^(@)` and the angle of depression from the top of another pole is `30^(@)` find the height of pole and distances of the point P from the poles.

Explanation

To solve the problem, we will follow these steps:

### Step 1: Understand the Problem
We have two poles of equal height on either side of a road that is 80 meters wide. From a point P between the poles, the angle of elevation to the top of one pole is \(60^\circ\) and the angle of depression to the top of the other pole is \(30^\circ\). We need to find the height of the poles and the distances from point P to each pole.

### Step 2: Define Variables
Let:
- \(h\) = height of the poles (in meters)
- \(x\) = distance from point P to the first pole (in meters)
- The distance from point P to the second pole will then be \(80 - x\) meters.

### Step 3: Set Up the Equations
Using the tangent of the angles, we can set up two equations based on the right triangles formed by the poles and point P.

1. For the pole with the angle of elevation of \(60^\circ\):
\[ \tan(60^\circ) = \frac{h}{x} \]
Since \(\tan(60^\circ) = \sqrt{3}\), we have:
\[ \sqrt{3} = \frac{h}{x} \implies h = \sqrt{3}x \quad \text{(Equation 1)} \]

2. For the pole with the angle of depression of \(30^\circ\):
\[ \tan(30^\circ) = \frac{h}{80 - x} \]
Since \(\tan(30^\circ) = \frac{1}{\sqrt{3}}\), we have:
\[ \frac{1}{\sqrt{3}} = \frac{h}{80 - x} \implies h = \frac{80 - x}{\sqrt{3}} \quad \text{(Equation 2)} \]

### Step 4: Equate the Two Expressions for Height
From Equation 1 and Equation 2, we can set them equal to each other:
\[ \sqrt{3}x = \frac{80 - x}{\sqrt{3}} \]

### Step 5: Solve for \(x\)
Multiply both sides by \(\sqrt{3}\) to eliminate the fraction:
\[ 3x = 80 - x \]
Now, add \(x\) to both sides:
\[ 3x + x = 80 \implies 4x = 80 \]
Divide both sides by 4:
\[ x = 20 \text{ meters} \]

### Step 6: Find the Height \(h\)
Now substitute \(x = 20\) back into Equation 1 to find \(h\):
\[ h = \sqrt{3} \cdot 20 = 20\sqrt{3} \text{ meters} \]

### Final Answers
- The height of each pole is \(20\sqrt{3}\) meters.
- The distance from point P to the first pole is \(20\) meters.
- The distance from point P to the second pole is \(80 - 20 = 60\) meters.