Two poles standing on horizontal ground are of heights 10 meters & 40 meters respectively. The line joining their tops makes an angle of `30^(@)` with the ground. The the distance (in meters) between the foot of the poles is

Options

A. 20
B. 30
C. `20sqrt3`
D. `30sqrt3`

Answer

To solve the problem, we need to find the distance between the foot of two poles of heights 10 meters and 40 meters, where the line joining their tops makes an angle of 30 degrees with the ground.

### Step-by-Step Solution:

1. **Identify the Poles and Their Heights**:
- Let the height of pole AB (the taller pole) be 40 meters.
- Let the height of pole CD (the shorter pole) be 10 meters.

2. **Define the Points**:
- Let point A be the top of pole AB, point B be the foot of pole AB.
- Let point C be the top of pole CD, point D be the foot of pole CD.
- The distance between the foot of the poles (BD) is what we need to find.

3. **Understanding the Geometry**:
- The line AC joining the tops of the poles makes an angle of 30 degrees with the ground.
- The vertical heights of the poles create two right triangles: triangle ABE and triangle CDE, where E is the point where the line AC meets the ground.

4. **Use Trigonometry in Triangle CDE**:
- In triangle CDE, we have:
- CD = 10 meters (the height of the shorter pole).
- DE is the horizontal distance from D to E.
- Using the tangent function:
\[ \tan(30^\circ) = \frac{CD}{DE} = \frac{10}{DE} \]
- Since \(\tan(30^\circ) = \frac{1}{\sqrt{3}}\), we can set up the equation:
\[ \frac{1}{\sqrt{3}} = \frac{10}{DE} \]
- Cross-multiplying gives:
\[ DE = 10\sqrt{3} \text{ meters} \]

5. **Use Trigonometry in Triangle ABE**:
- In triangle ABE, we have:
- AB = 40 meters (the height of the taller pole).
- BE is the horizontal distance from B to E, which is what we need to find (let's call it \(x\)).
- Using the tangent function again:
\[ \tan(30^\circ) = \frac{AB}{BE} = \frac{40}{x + DE} \]
- Again, since \(\tan(30^\circ) = \frac{1}{\sqrt{3}}\), we can set up the equation:
\[ \frac{1}{\sqrt{3}} = \frac{40}{x + 10\sqrt{3}} \]
- Cross-multiplying gives:
\[ x + 10\sqrt{3} = 40\sqrt{3} \]

6. **Solve for \(x\)**:
- Rearranging the equation:
\[ x = 40\sqrt{3} - 10\sqrt{3} \]
- Simplifying gives:
\[ x = 30\sqrt{3} \text{ meters} \]

### Final Answer:
The distance between the foot of the poles is \(30\sqrt{3}\) meters.