Two vertical poles of different heights are standing 20m away from each other on the level ground. The angle of elevation of the top of the first pole from the foot of the second pole is `60^(@)` and angle of elevation of the top of the second pole from the foot of the first pole is`30^(@)`. Find the difference between the heights of two poles. (Take`sqrt(3)=1.73`)

Options

A. `23.06`
B. `26`
C. `24`
D. None

Explanation

To solve the problem of finding the difference between the heights of two vertical poles given the angles of elevation from each pole, we can follow these steps:

### Step 1: Understand the Setup
We have two poles, A and B, standing 20 meters apart. The angle of elevation from the foot of pole B to the top of pole A is \(60^\circ\), and the angle of elevation from the foot of pole A to the top of pole B is \(30^\circ\).

### Step 2: Identify the Triangles
We can form two right triangles:
1. Triangle ACB where:
- AC is the height of pole A (let's denote it as \(h_A\)),
- BC is the height of pole B (let's denote it as \(h_B\)),
- AB is the distance between the poles, which is 20 m.

2. Triangle BCD where:
- Angle CBA = \(30^\circ\) (from pole A to pole B),
- Angle ACB = \(60^\circ\) (from pole B to pole A).

### Step 3: Use Trigonometric Ratios
Using the tangent function, we can express the heights of the poles in terms of the distances and angles.

#### For Pole A (from Pole B):
Using triangle BCA:
\[ \tan(60^\circ) = \frac{h_A}{20} \]
Since \(\tan(60^\circ) = \sqrt{3}\):
\[ \sqrt{3} = \frac{h_A}{20} \]
Thus,
\[ h_A = 20\sqrt{3} \]

#### For Pole B (from Pole A):
Using triangle ABC:
\[ \tan(30^\circ) = \frac{h_B}{20} \]
Since \(\tan(30^\circ) = \frac{1}{\sqrt{3}}\):
\[ \frac{1}{\sqrt{3}} = \frac{h_B}{20} \]
Thus,
\[ h_B = \frac{20}{\sqrt{3}} = \frac{20\sqrt{3}}{3} \]

### Step 4: Calculate the Heights
Now we have:
- Height of pole A: \(h_A = 20\sqrt{3}\)
- Height of pole B: \(h_B = \frac{20\sqrt{3}}{3}\)

### Step 5: Find the Difference in Heights
To find the difference in heights:
\[ \text{Difference} = h_A - h_B = 20\sqrt{3} - \frac{20\sqrt{3}}{3} \]

### Step 6: Simplify the Expression
To simplify:
\[ \text{Difference} = 20\sqrt{3} \left(1 - \frac{1}{3}\right) = 20\sqrt{3} \cdot \frac{2}{3} = \frac{40\sqrt{3}}{3} \]

### Step 7: Substitute \(\sqrt{3}\)
Using \(\sqrt{3} \approx 1.73\):
\[ \text{Difference} = \frac{40 \times 1.73}{3} = \frac{69.2}{3} \approx 23.07 \text{ meters} \]

### Final Answer
The difference between the heights of the two poles is approximately **23.07 meters**.