From a point 30m from the foot of a tree, the angle of elevation of the top of the tree is ` 30^(@)` . What is the height of the tree ?

Options

A. 10
B. `10 sqrt"" 2 `
C. ` 10 sqrt""3 `
D. None of these

Answer

To find the height of the tree given the distance from the foot of the tree and the angle of elevation, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Problem**:
- We have a tree and a point from which the angle of elevation to the top of the tree is given.
- The distance from the point to the foot of the tree is 30 meters, and the angle of elevation is \(30^\circ\).

2. **Draw a Diagram**:
- Let point A be the top of the tree, point B be the foot of the tree, and point C be the point from which the angle is measured.
- The distance BC (horizontal distance from the point to the foot of the tree) is 30 meters.
- The angle of elevation \( \angle ACB = 30^\circ \).

3. **Use the Tangent Function**:
- The tangent of an angle in a right triangle is defined as the ratio of the opposite side (height of the tree, AB) to the adjacent side (distance from the point to the foot of the tree, BC).
- Therefore, we can write:
\[ \tan(30^\circ) = \frac{AB}{BC} \]

4. **Substitute Known Values**:
- We know that \( BC = 30 \) meters and \( \tan(30^\circ) = \frac{1}{\sqrt{3}} \).
- Substituting these values into the equation gives:
\[ \frac{1}{\sqrt{3}} = \frac{AB}{30} \]

5. **Solve for AB (Height of the Tree)**:
- Rearranging the equation to find AB:
\[ AB = 30 \cdot \tan(30^\circ) \]
- Substituting the value of \( \tan(30^\circ) \):
\[ AB = 30 \cdot \frac{1}{\sqrt{3}} \]
- Simplifying this gives:
\[ AB = \frac{30}{\sqrt{3}} = 10\sqrt{3} \]

6. **Final Answer**:
- The height of the tree is \( 10\sqrt{3} \) meters.