A straight tree breaks due to storm and the broken part bends so that the top of the tree touches the ground making an angle of `30^(@)` with the ground .The distance from the foot of the tree to the point , where the top touches the ground is 10 m . Find the total height of the tree ?

Options

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

Answer

To solve the problem step by step, let's break it down:

### Step 1: Understand the Problem
We have a tree that has broken due to a storm. The broken part bends down to touch the ground, forming an angle of 30 degrees with the ground. The distance from the base of the tree to the point where the top touches the ground is given as 10 meters. We need to find the total height of the tree.

### Step 2: Visualize the Situation
Let's denote:
- Point A as the top of the tree before it broke.
- Point B as the point where the tree is rooted (the base).
- Point C as the point where the tree broke.
- Point D as the point where the top of the tree touches the ground.

The angle ∠BDC is 30 degrees, and BD (the distance from the base to the point where the top touches the ground) is 10 meters.

### Step 3: Use Trigonometry
In triangle BDC, we can use the tangent of angle ∠BDC to find the height of the broken part of the tree (CD).

Using the tangent function:
\[ \tan(30^\circ) = \frac{\text{opposite}}{\text{adjacent}} = \frac{CD}{BD} \]
Where:
- CD is the height of the broken part of the tree.
- BD is 10 meters.

### Step 4: Calculate CD
We know that:
\[ \tan(30^\circ) = \frac{1}{\sqrt{3}} \]
Substituting the values into the equation:
\[ \frac{1}{\sqrt{3}} = \frac{CD}{10} \]
Now, we can solve for CD:
\[ CD = 10 \cdot \frac{1}{\sqrt{3}} = \frac{10}{\sqrt{3}} \text{ meters} \]

### Step 5: Find the Total Height of the Tree
The total height of the tree (AB) is the sum of the height from the base to the break point (BC) and the height of the broken part (CD). Since the broken part (CD) is equal to the height from the break point to the top (AC), we have:
\[ AB = BC + CD \]
Since BC = CD (because the tree broke and the top part is equal to the height above the break), we can write:
\[ AB = CD + CD = 2 \cdot CD \]
Substituting the value of CD:
\[ AB = 2 \cdot \frac{10}{\sqrt{3}} = \frac{20}{\sqrt{3}} \text{ meters} \]

### Step 6: Rationalize the Denominator
To express this in a more standard form, we can rationalize the denominator:
\[ AB = \frac{20}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}} = \frac{20\sqrt{3}}{3} \text{ meters} \]

### Step 7: Conclusion
Thus, the total height of the tree is:
\[ \text{Total Height} = \frac{20\sqrt{3}}{3} \text{ meters} \]