There is a small island in the middle of a 100 m wide river and a tall tree stands on the island. P and Q are points directly opposite to each other on two banks and in line with the tree. If the angles of elevation of the top of the tree from P and Q are respectively `30^(@)` and `45^(@)` find the height of the tree

Options

A. `35.3` m
B. `35.6` cm
C. 35 m
D. `36.60` m

Answer

To find the height of the tree on the island, we can use trigonometric ratios based on the angles of elevation from points P and Q.

### Step-by-Step Solution:

1. **Understanding the Setup**:
- Let the height of the tree be \( H \).
- The distance from point P to the base of the tree (on the island) is \( x \).
- The distance from point Q to the base of the tree is \( 100 - x \) (since the river is 100 m wide).

2. **Using the Angle of Elevation from Point P**:
- The angle of elevation from point P to the top of the tree is \( 30^\circ \).
- Using the tangent function:
\[ \tan(30^\circ) = \frac{H}{x} \]
- We know that \( \tan(30^\circ) = \frac{1}{\sqrt{3}} \), so:
\[ \frac{1}{\sqrt{3}} = \frac{H}{x} \]
- Rearranging gives:
\[ H = \frac{x}{\sqrt{3}} \quad \text{(Equation 1)} \]

3. **Using the Angle of Elevation from Point Q**:
- The angle of elevation from point Q to the top of the tree is \( 45^\circ \).
- Again using the tangent function:
\[ \tan(45^\circ) = \frac{H}{100 - x} \]
- Since \( \tan(45^\circ) = 1 \), we have:
\[ 1 = \frac{H}{100 - x} \]
- Rearranging gives:
\[ H = 100 - x \quad \text{(Equation 2)} \]

4. **Setting the Equations Equal**:
- From Equation 1 and Equation 2, we can set them equal to each other:
\[ \frac{x}{\sqrt{3}} = 100 - x \]

5. **Solving for \( x \)**:
- Multiply both sides by \( \sqrt{3} \):
\[ x = \sqrt{3}(100 - x) \]
- Expanding gives:
\[ x = 100\sqrt{3} - \sqrt{3}x \]
- Rearranging terms:
\[ x + \sqrt{3}x = 100\sqrt{3} \]
\[ x(1 + \sqrt{3}) = 100\sqrt{3} \]
- Solving for \( x \):
\[ x = \frac{100\sqrt{3}}{1 + \sqrt{3}} \]

6. **Finding \( H \)**:
- Substitute \( x \) back into either Equation 1 or Equation 2 to find \( H \). Using Equation 2:
\[ H = 100 - x = 100 - \frac{100\sqrt{3}}{1 + \sqrt{3}} \]
- Simplifying this expression:
\[ H = 100 - \frac{100\sqrt{3}}{1 + \sqrt{3}} = \frac{100(1 + \sqrt{3}) - 100\sqrt{3}}{1 + \sqrt{3}} = \frac{100}{1 + \sqrt{3}} \]

7. **Final Calculation**:
- To find the approximate value of \( H \):
\[ H \approx 36.6 \text{ m} \]

### Final Answer:
The height of the tree is approximately **36.6 meters**.