The angle of depression of a point on the ground from the top of a tree is `60^(@)`. Lowering down 20 m from the top, the angle of depression changes to `30^(@)`. Find the distance between the point and the foot of the tree. Also find the height of the tree.

Answer

To solve the problem, we will use the concepts of trigonometry, particularly the tangent function, which relates angles to the ratios of the sides of right triangles.

### Step-by-Step Solution:

1. **Understanding the Problem**:
- Let the height of the tree be \( h \).
- From the top of the tree, the angle of depression to a point on the ground is \( 60^\circ \).
- When we lower down 20 m from the top of the tree, the angle of depression changes to \( 30^\circ \).

2. **Setting Up the Diagram**:
- Let \( A \) be the top of the tree, \( B \) be the point 20 m below \( A \), and \( C \) be the point on the ground directly below \( A \).
- Let \( D \) be the point on the ground directly below \( B \).
- The distance from the foot of the tree (point \( C \)) to the point on the ground (point \( D \)) is what we need to find.

3. **Using the Angle of Depression**:
- From point \( A \) (top of the tree), the angle of depression to point \( D \) is \( 60^\circ \).
- From point \( B \) (20 m lower), the angle of depression to point \( D \) is \( 30^\circ \).

4. **Applying Trigonometry**:
- In triangle \( ACD \):
\[ \tan(60^\circ) = \frac{h}{CD} \]
\[ \sqrt{3} = \frac{h}{CD} \quad \Rightarrow \quad h = CD \cdot \sqrt{3} \quad \text{(1)} \]

- In triangle \( BCD \):
\[ \tan(30^\circ) = \frac{h - 20}{CD} \]
\[ \frac{1}{\sqrt{3}} = \frac{h - 20}{CD} \quad \Rightarrow \quad h - 20 = \frac{CD}{\sqrt{3}} \quad \Rightarrow \quad h = \frac{CD}{\sqrt{3}} + 20 \quad \text{(2)} \]

5. **Equating the Two Expressions for \( h \)**:
- From (1) and (2):
\[ CD \cdot \sqrt{3} = \frac{CD}{\sqrt{3}} + 20 \]
- Multiply through by \( \sqrt{3} \) to eliminate the fraction:
\[ 3CD = CD + 20\sqrt{3} \]
- Rearranging gives:
\[ 2CD = 20\sqrt{3} \]
\[ CD = 10\sqrt{3} \quad \text{(3)} \]

6. **Finding the Height of the Tree**:
- Substitute \( CD \) back into equation (1):
\[ h = 10\sqrt{3} \cdot \sqrt{3} = 30 \quad \text{(height of the tree)} \]

7. **Final Answers**:
- The distance from the foot of the tree to the point on the ground is \( CD = 10\sqrt{3} \).
- The height of the tree is \( h = 30 \) m.

### Summary:
- **Distance from the foot of the tree to the point on the ground**: \( 10\sqrt{3} \) m
- **Height of the tree**: \( 30 \) m