The upper part of a tree broken at a certain height makes an angle of `60^(@)` with the ground at a distance of 10 metre from its foot The original height of the tree was

Options

A. `20sqrt(3)` metre
B. `10sqrt(3)` metre
C. `10(2+sqrt(3))` metre
D. `10(2-sqrt(3))` metre

Answer

To find the original height of the tree, we can break down the problem step by step.

### Step 1: Understand the Problem
We have a tree that has broken at a certain height. The broken part makes an angle of \(60^\circ\) with the ground at a distance of 10 meters from the foot of the tree. We need to find the original height of the tree.

### Step 2: Visualize the Situation
Let:
- \(A\) be the point where the tree was originally standing.
- \(B\) be the point where the top of the tree (point \(C\)) has fallen to the ground.
- \(C\) be the original top of the tree.

The distance from the foot of the tree (point \(A\)) to the point where the top of the tree has fallen (point \(B\)) is 10 meters. The angle \(CAB\) (the angle between the line \(AC\) and the ground) is \(60^\circ\).

### Step 3: Use Trigonometry to Find \(AB\)
Using the tangent function, we can express the height \(AB\) (the height of the broken part of the tree) in terms of the distance \(AB\) and the angle \(60^\circ\):

\[ \tan(60^\circ) = \frac{AB}{10} \]

Since \(\tan(60^\circ) = \sqrt{3}\), we have:

\[ \sqrt{3} = \frac{AB}{10} \]

From this, we can solve for \(AB\):

\[ AB = 10\sqrt{3} \]

### Step 4: Find the Length of \(BC\)
Now, we need to find the length of \(BC\) (the height of the part of the tree that is still standing). We can use the Pythagorean theorem in triangle \(ABC\):

\[ AC^2 = AB^2 + BC^2 \]

Here, \(AC\) is the original height of the tree, which we can denote as \(h\). Thus, we have:

\[ h^2 = (10\sqrt{3})^2 + BC^2 \]

Calculating \( (10\sqrt{3})^2 \):

\[ (10\sqrt{3})^2 = 100 \times 3 = 300 \]

So, we have:

\[ h^2 = 300 + BC^2 \]

### Step 5: Find \(BC\) Using the Pythagorean Theorem
We know that \(AB\) is the height of the broken part, and we can find \(BC\) using the right triangle formed by the height of the tree and the distance from the foot of the tree:

Using the Pythagorean theorem again, we can express \(BC\) as:

\[ BC = \sqrt{(10)^2 + (10\sqrt{3})^2} = \sqrt{100 + 300} = \sqrt{400} = 20 \]

### Step 6: Calculate the Original Height of the Tree
Now we can find the original height \(h\) of the tree:

\[ h = AB + BC = 10\sqrt{3} + 20 \]

### Final Answer
Thus, the original height of the tree is:

\[ h = 10\sqrt{3} + 20 \text{ meters} \]