Two poles of height 10 meters and 20 meters stand at the centres of two circular plots which touch each other externally at a point and the two poles subtend angles `30^(@) and 60^(@)` respectively at this point, then the distance between the centres of these circular plots is

Options

A. 30 meters
B. `(50)/(sqrt3)` meters
C. `(70)/(sqrt3)` meters
D. `(10sqrt3+20)` meters

Answer

To solve the problem, we need to find the distance between the centers of the two circular plots based on the heights of the poles and the angles they subtend at the point where the circles touch externally.

### Step-by-Step Solution:

1. **Identify the Given Information**:
- Height of the first pole (h1) = 10 meters
- Height of the second pole (h2) = 20 meters
- Angle subtended by the first pole (θ1) = 30 degrees
- Angle subtended by the second pole (θ2) = 60 degrees

2. **Use the Cotangent Function**:
The distance from the base of each pole to the point where the circles touch can be calculated using the cotangent of the angles:
- For the first pole:
\[ OA = h_1 \cdot \cot(\theta_1) = 10 \cdot \cot(30^\circ) \]
- For the second pole:
\[ OB = h_2 \cdot \cot(\theta_2) = 20 \cdot \cot(60^\circ) \]

3. **Calculate the Cotangent Values**:
- We know that:
\[ \cot(30^\circ) = \sqrt{3} \quad \text{and} \quad \cot(60^\circ) = \frac{1}{\sqrt{3}} \]
- Substitute these values into the equations:
\[ OA = 10 \cdot \sqrt{3} \]
\[ OB = 20 \cdot \frac{1}{\sqrt{3}} = \frac{20}{\sqrt{3}} \]

4. **Find the Total Distance Between the Centers**:
The total distance \( D \) between the centers of the circular plots is given by:
\[ D = OA + OB = 10\sqrt{3} + \frac{20}{\sqrt{3}} \]

5. **Combine the Terms**:
To combine these terms, we can express \( 10\sqrt{3} \) with a common denominator:
\[ D = 10\sqrt{3} + \frac{20}{\sqrt{3}} = \frac{10\sqrt{3} \cdot \sqrt{3}}{\sqrt{3}} + \frac{20}{\sqrt{3}} = \frac{30 + 20}{\sqrt{3}} = \frac{50}{\sqrt{3}} \]

6. **Rationalize the Denominator**:
To express the final answer without a square root in the denominator:
\[ D = \frac{50}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}} = \frac{50\sqrt{3}}{3} \]

### Final Answer:
The distance between the centers of the circular plots is:
\[ D = \frac{50\sqrt{3}}{3} \text{ meters} \]