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. `5(sqrt3+1)`
B. `(5)/(2)(2+sqrt(3))`
C. `10(sqrt(3)-1)`
D. `5(2+sqrt(3))`

Explanation

To find the distance between the two poles, we can follow these steps:

### Step 1: Understand the problem and draw a diagram
We have two poles, one of height 5 m and the other of height 10 m. The line joining the tops of the two poles makes an angle of 15 degrees with the ground. Let's denote the points as follows:
- Point A: Top of the 5 m pole
- Point B: Top of the 10 m pole
- Point C: Base of the 5 m pole
- Point D: Base of the 10 m pole

### Step 2: Set up the right triangle
We can form a right triangle with the following points:
- Point E: The foot of the vertical line from point B to the ground level (which is the same horizontal level as points C and D).
- The height from point E to point B is 10 m, and the height from point E to point A is 5 m.

### Step 3: Determine the vertical distance between the tops of the poles
The vertical distance (height difference) between the tops of the poles A and B is:
\[ AB = 10 \, \text{m} - 5 \, \text{m} = 5 \, \text{m} \]

### Step 4: Use trigonometry to find the horizontal distance
In triangle EAB, we know:
- The opposite side (height difference) = 5 m
- The angle with the ground = 15 degrees

Using the tangent function, we have:
\[ \tan(15^\circ) = \frac{\text{opposite}}{\text{adjacent}} = \frac{5}{d} \]
where \(d\) is the horizontal distance between the foot of the poles (C and D).

### Step 5: Rearranging the equation
Rearranging the equation gives:
\[ d = \frac{5}{\tan(15^\circ)} \]

### Step 6: Calculate \(d\)
Using the value of \(\tan(15^\circ)\):
\[ \tan(15^\circ) = 2 - \sqrt{3} \approx 0.268 \]
Thus,
\[ d = \frac{5}{0.268} \approx 18.66 \, \text{m} \]

### Step 7: Conclusion
The distance between the two poles is approximately \(18.66 \, \text{m}\).

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