The top of a broken tree has its top end touching the ground at a distance 15 m from the bottom, the angle made by the broken end with the ground is `30^@`. Then length of broken part is

Options

A. 10 m
B. `sqrt3 m`
C. `5sqrt3 m`
D. `10sqrt3m`

Answer

To find the length of the broken part of the tree, we can follow these steps:

### Step 1: Understand the Geometry of the Problem
We have a broken tree represented as a right triangle. The point where the tree breaks is point A, the point where the tree touches the ground is point M, and the base of the tree is point B. The distance from point B to point M is given as 15 m, and the angle made by the broken part (AM) with the ground (BM) is 30 degrees.

### Step 2: Identify the Right Triangle
In the right triangle ABM:
- AM is the hypotenuse (the length of the broken part).
- BM is the base (the distance from the bottom of the tree to the point where the top touches the ground), which is 15 m.
- Angle AMB is 30 degrees.

### Step 3: Use Trigonometric Ratios
We can use the cosine function to relate the adjacent side (BM) and the hypotenuse (AM):
\[ \cos(\theta) = \frac{\text{Adjacent}}{\text{Hypotenuse}} \]
Here, \(\theta = 30^\circ\), the adjacent side is BM (15 m), and the hypotenuse is AM (which we want to find).

### Step 4: Substitute Values into the Cosine Formula
\[ \cos(30^\circ) = \frac{BM}{AM} \]
Substituting the known values:
\[ \cos(30^\circ) = \frac{15}{AM} \]
We know that \(\cos(30^\circ) = \frac{\sqrt{3}}{2}\), so we can write:
\[ \frac{\sqrt{3}}{2} = \frac{15}{AM} \]

### Step 5: Solve for AM
Cross-multiplying gives:
\[ AM \cdot \sqrt{3} = 30 \]
Now, divide both sides by \(\sqrt{3}\):
\[ AM = \frac{30}{\sqrt{3}} = 10\sqrt{3} \text{ m} \]

### Conclusion
The length of the broken part of the tree is \(10\sqrt{3}\) meters.

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