If a flag-staff of 6 m height plated on the top of a tower throws a shadow of `2 sqrt3` m along the ground then what is the angle that the sun makes with the ground ?

Options

A. `60^(@)`
B. `45 ^(@)`
C. `30^(@)`
D. `15^(@)`

Answer

To solve the problem, we need to find the angle that the sun makes with the ground based on the height of the flagstaff and the length of its shadow. Let's break down the solution step by step.

### Step-by-Step Solution:

1. **Understand the Problem**:
We have a flagstaff of height \(6 \, \text{m}\) placed on top of a tower. The shadow of the flagstaff is given as \(2\sqrt{3} \, \text{m}\). We need to find the angle \(\theta\) that the sun makes with the ground.

2. **Identify the Components**:
- Height of the flagstaff = \(6 \, \text{m}\)
- Length of the shadow = \(2\sqrt{3} \, \text{m}\)
- Let the height of the tower be \(h\).

3. **Set Up the Right Triangle**:
In the right triangle formed by the flagstaff and its shadow:
- The opposite side (height of the flagstaff) = \(6 \, \text{m}\)
- The adjacent side (length of the shadow) = \(2\sqrt{3} \, \text{m}\)

4. **Use the Tangent Function**:
The tangent of the angle \(\theta\) can be expressed as:
\[ \tan(\theta) = \frac{\text{Opposite}}{\text{Adjacent}} = \frac{6}{2\sqrt{3}} \]

5. **Simplify the Expression**:
\[ \tan(\theta) = \frac{6}{2\sqrt{3}} = \frac{3}{\sqrt{3}} = \sqrt{3} \]

6. **Find the Angle**:
We know that:
\[ \tan(60^\circ) = \sqrt{3} \]
Therefore, \(\theta = 60^\circ\).

7. **Conclusion**:
The angle that the sun makes with the ground is \(60^\circ\).