What should be the height of a flag where a 20 feet long ladder reaches 20 feet below the flag (The angle of elevationof the top of the flag at the foot of the ladder is `60^(@)`?

Options

A. 20 feet
B. 30 feet
C. 40 feet
D. `20 sqrt2 ` feet

Answer

To solve the problem step by step, we will analyze the situation involving the ladder, the flag, and the angles involved.

### Step-by-Step Solution:

1. **Understanding the Problem**:
- We have a flag at a certain height, and a 20-foot long ladder reaches 20 feet below the flag.
- The angle of elevation to the top of the flag from the foot of the ladder is 60 degrees.

2. **Setting Up the Diagram**:
- Let the height of the flag be represented as \( AB \).
- The point where the ladder touches the ground is point \( D \).
- The point where the ladder touches the wall (the flag) is point \( C \).
- The height from point \( D \) to point \( B \) (the top of the flag) is \( h \).
- Since the ladder reaches 20 feet below the flag, the height from point \( D \) to point \( A \) (the bottom of the flag) is \( 20 \) feet.

3. **Identifying the Heights**:
- Therefore, we can express the height of the flag \( AB \) as:
\[ AB = AD + DB = 20 + h \]

4. **Using Trigonometry**:
- In triangle \( DBC \), we can use the sine function since we know the angle of elevation and the opposite side.
- The angle of elevation \( \angle DCB = 60^\circ \).
- The length \( DC \) (the height from point \( D \) to point \( C \)) is the same as \( h \), and \( DB \) (the length of the ladder) is 20 feet.

5. **Applying the Sine Function**:
- Using the sine of the angle:
\[ \sin(60^\circ) = \frac{h}{20} \]
- We know that \( \sin(60^\circ) = \frac{\sqrt{3}}{2} \), so we can write:
\[ \frac{\sqrt{3}}{2} = \frac{h}{20} \]

6. **Solving for \( h \)**:
- Rearranging the equation gives:
\[ h = 20 \cdot \frac{\sqrt{3}}{2} = 10\sqrt{3} \]

7. **Calculating the Total Height of the Flag**:
- Now substituting \( h \) back into the height of the flag:
\[ AB = 20 + 10\sqrt{3} \]

8. **Final Calculation**:
- We can approximate \( \sqrt{3} \approx 1.732 \):
\[ h \approx 10 \cdot 1.732 = 17.32 \text{ feet} \]
- Thus, the height of the flag is approximately:
\[ AB \approx 20 + 17.32 \approx 37.32 \text{ feet} \]

### Conclusion:
The height of the flag is approximately 37.32 feet.