Two poles standing on a horizontal ground are of heights 5 m and 10 m, respectively. The line joining their tops makes an angle of `15^(@)` with the ground. Then, the distance (in m) between the poles, is

Options

A. `10 (sqrt(3) - 1)`
B. `5 (sqrt(3) + 1)`
C. `(5)/(2) (2 + sqrt(3))`
D. `5 (2 + sqrt(3))`

Explanation

To solve the problem of finding the distance between two poles of heights 5 m and 10 m, with the line joining their tops making an angle of \(15^\circ\) with the ground, we can follow these steps:

### Step 1: Understand the Geometry
Let:
- Point A be the top of the first pole (height 5 m).
- Point B be the top of the second pole (height 10 m).
- Point C be the base of the first pole.
- Point D be the base of the second pole.

The heights of the poles give us the vertical distances:
- AC = 5 m (height of the first pole)
- BD = 10 m (height of the second pole)

### Step 2: Set Up the Right Triangle
The line joining the tops of the poles (AB) makes an angle of \(15^\circ\) with the ground. We can denote the distance between the bases of the poles (CD) as \(x\).

### Step 3: Identify the Vertical Difference
The vertical difference between the tops of the poles is:
\[ BD - AC = 10 - 5 = 5 \text{ m} \]

### Step 4: Use Trigonometry
In triangle ADB, we can use the tangent function, which relates the opposite side to the adjacent side:
\[ \tan(15^\circ) = \frac{\text{opposite}}{\text{adjacent}} = \frac{BD - AC}{CD} \]
Substituting the known values:
\[ \tan(15^\circ) = \frac{5}{x} \]

### Step 5: Solve for \(x\)
Using the known value of \(\tan(15^\circ)\):
\[ \tan(15^\circ) = 2 - \sqrt{3} \]
We can set up the equation:
\[ 2 - \sqrt{3} = \frac{5}{x} \]
Rearranging gives:
\[ x(2 - \sqrt{3}) = 5 \]
Thus:
\[ x = \frac{5}{2 - \sqrt{3}} \]

### Step 6: Rationalize the Denominator
To rationalize the denominator:
\[ x = \frac{5}{2 - \sqrt{3}} \cdot \frac{2 + \sqrt{3}}{2 + \sqrt{3}} = \frac{5(2 + \sqrt{3})}{(2 - \sqrt{3})(2 + \sqrt{3})} \]
Calculating the denominator:
\[ (2 - \sqrt{3})(2 + \sqrt{3}) = 4 - 3 = 1 \]
So we have:
\[ x = 5(2 + \sqrt{3}) \]

### Final Answer
The distance between the two poles is:
\[ \boxed{5(2 + \sqrt{3})} \text{ m} \]