A pole broken by the storm of wind and its top struck the ground at the angle of `30^(@)` and at a distance of 20 m from the foot of the pole. The height of the pole before it was broken was

Options

A. `20sqrt(3)m`
B. `(40sqrt(3))/3` m
C. `60sqrt(3)` m
D. `(100sqrt(3))/3` m

Answer

To find the height of the pole before it was broken, we can follow these steps:

### Step 1: Understand the Problem
We have a pole that has been broken by a storm. The top of the pole has struck the ground at an angle of \(30^\circ\) and is \(20\) meters away from the foot of the pole. We need to find the height of the pole before it was broken.

### Step 2: Draw a Diagram
Draw a right triangle where:
- Point O is the foot of the pole.
- Point A is the point where the pole was broken.
- Point B is the point where the top of the pole (point A) touches the ground.
- Point C is the point directly above O at the height of the pole before it was broken.

### Step 3: Identify the Triangle
In triangle OAB:
- Angle AOB = \(30^\circ\)
- Distance OB (the distance from the foot of the pole to the point where the top struck the ground) = \(20\) m.

### Step 4: Use Trigonometric Ratios
Using the tangent of angle \(30^\circ\):
\[ \tan(30^\circ) = \frac{AB}{OB} \]
Where:
- \(AB\) is the height of the pole above point A (the broken part).
- \(OB = 20\) m.

Since \(\tan(30^\circ) = \frac{1}{\sqrt{3}}\), we can write:
\[ \frac{AB}{20} = \frac{1}{\sqrt{3}} \]
Thus:
\[ AB = \frac{20}{\sqrt{3}} \text{ m} \]

### Step 5: Find the Length of OA
Let \(OA\) be the length of the pole above point A (the broken part). Since triangle OAC is also a right triangle, we can use the properties of the triangle:
- \(OA\) is the same as \(AC\) (the height of the pole before it was broken).

Using the sine of angle \(30^\circ\):
\[ \sin(30^\circ) = \frac{AC}{OB} \]
Where:
- \(AC = OA\)
- \(OB = 20\) m.

Since \(\sin(30^\circ) = \frac{1}{2}\), we can write:
\[ \frac{AC}{20} = \frac{1}{2} \]
Thus:
\[ AC = 10 \text{ m} \]

### Step 6: Calculate the Total Height of the Pole
The total height of the pole before it was broken is:
\[ Height = OA + AB = AC + AB = 10 + \frac{20}{\sqrt{3}} \text{ m} \]

### Step 7: Rationalize the Height
To find the total height in a simplified form:
\[ Height = 10 + \frac{20}{\sqrt{3}} = 10 + \frac{20\sqrt{3}}{3} \]
Thus, the height of the pole before it was broken is:
\[ Height = 10 + \frac{20\sqrt{3}}{3} \text{ m} \]

### Final Result
The height of the pole before it was broken is:
\[ \text{Height} = 20\sqrt{3} \text{ m} \]

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