A tree is broken by the wind. If the top of the tree struck the ground at an angle of `30^(@)` and at a distance of 30 m from the root, then the height of the tree is

Options

A. `25 sqrt3` m
B. `30sqrt3` m
C. `15 sqrt3` m
D. `20 sqrt3` m

Answer

To find the height of the tree that was broken by the wind, we can use trigonometric relationships and the Pythagorean theorem. Here’s a step-by-step solution:

### Step 1: Understand the scenario
The tree breaks and the top of the tree strikes the ground at an angle of \(30^\circ\) from the horizontal. The distance from the base of the tree to the point where the top of the tree touches the ground is 30 meters.

### Step 2: Set up the triangle
When the tree breaks, it forms a right triangle where:
- The height of the tree (before it broke) can be represented as \(x + y\), where \(x\) is the height of the broken part and \(y\) is the height of the part that remains standing.
- The distance from the base of the tree to the point where the top touches the ground is the base of the triangle (30 m).
- The angle between the ground and the broken part of the tree is \(30^\circ\).

### Step 3: Use trigonometric ratios
In a right triangle, the tangent of an angle is the ratio of the opposite side (height of the tree) to the adjacent side (distance from the base). Therefore, we can write:
\[ \tan(30^\circ) = \frac{y}{30} \]
We know that \(\tan(30^\circ) = \frac{1}{\sqrt{3}}\). Thus:
\[ \frac{1}{\sqrt{3}} = \frac{y}{30} \]

### Step 4: Solve for \(y\)
Rearranging the equation gives:
\[ y = 30 \cdot \frac{1}{\sqrt{3}} = \frac{30}{\sqrt{3}} = 10\sqrt{3} \text{ meters} \]

### Step 5: Find \(x\) using the Pythagorean theorem
Now, we can find \(x\) using the Pythagorean theorem. In our triangle:
\[ x^2 + y^2 = (30)^2 \]
Substituting \(y = 10\sqrt{3}\):
\[ x^2 + (10\sqrt{3})^2 = 900 \]
\[ x^2 + 300 = 900 \]
\[ x^2 = 600 \]
\[ x = \sqrt{600} = 10\sqrt{6} \text{ meters} \]

### Step 6: Calculate the total height of the tree
The total height of the tree before it broke is:
\[ \text{Height} = x + y = 10\sqrt{6} + 10\sqrt{3} \]

### Step 7: Final answer
Thus, the height of the tree is:
\[ \text{Height} = 10(\sqrt{6} + \sqrt{3}) \text{ meters} \]