The angle of elevation ofthe top of a tree from a point on the ground which is 300 m away from thetree is `30^(@)` When the tree grew up, its angle ofelevation ofthe top of it became `60^(@)` from the same point. How much did the tree grow? (nearest to an integer)

Options

A. 342 m
B. 364 m
C. 384 m
D. 346 m

Answer

To solve the problem, we need to find out how much the tree grew after its angle of elevation changed from \(30^\circ\) to \(60^\circ\). We will use trigonometric relationships to find the heights of the tree at both angles and then calculate the difference.

### Step-by-Step Solution:

1. **Identify the given information**:
- Distance from the point on the ground to the base of the tree = 300 m
- Angle of elevation when the tree was shorter = \(30^\circ\)
- Angle of elevation when the tree grew taller = \(60^\circ\)

2. **Set up the first equation using the angle of elevation of \(30^\circ\)**:
- Let \(x\) be the height of the tree when the angle of elevation was \(30^\circ\).
- Using the tangent function:
\[ \tan(30^\circ) = \frac{x}{300} \]
- We know that \(\tan(30^\circ) = \frac{1}{\sqrt{3}}\):
\[ \frac{1}{\sqrt{3}} = \frac{x}{300} \]
- Rearranging gives:
\[ x = 300 \cdot \frac{1}{\sqrt{3}} = \frac{300}{\sqrt{3}} = 100\sqrt{3} \text{ m} \]

3. **Set up the second equation using the angle of elevation of \(60^\circ\)**:
- Let \(y\) be the height of the tree when the angle of elevation is \(60^\circ\).
- Again using the tangent function:
\[ \tan(60^\circ) = \frac{y}{300} \]
- We know that \(\tan(60^\circ) = \sqrt{3}\):
\[ \sqrt{3} = \frac{y}{300} \]
- Rearranging gives:
\[ y = 300\sqrt{3} \text{ m} \]

4. **Calculate the growth of the tree**:
- The growth of the tree is the difference between the two heights:
\[ \text{Growth} = y - x = 300\sqrt{3} - 100\sqrt{3} = 200\sqrt{3} \text{ m} \]

5. **Approximate the growth to the nearest integer**:
- Using \(\sqrt{3} \approx 1.732\):
\[ \text{Growth} \approx 200 \cdot 1.732 = 346.4 \text{ m} \]
- Rounding to the nearest integer gives:
\[ \text{Growth} \approx 346 \text{ m} \]

### Final Answer:
The tree grew approximately **346 m**.