If a flag-staff 6 metres high placed o the top of a tower throws a shadow of `2sqrt(3)`m along the ground, then the angle (in degrees) that the sun makes with the ground is

Options

A. `60^(@)`
B. `30^(@)`
C. `45^(@)`
D. None

Answer

To solve the problem step by step, we will use trigonometric ratios to find the angle of elevation of the sun based on the height of the flagstaff and the length of the shadow.

### Step-by-Step Solution:

1. **Understand the Problem**:
We have a flagstaff that is 6 meters high, and it is placed on top of a tower. The shadow of the flagstaff on the ground is `2√3` meters long. We need to find the angle of elevation of the sun.

2. **Identify the Right Triangle**:
The situation can be visualized as a right triangle where:
- The height of the flagstaff (6 meters) is the perpendicular side.
- The length of the shadow (`2√3` meters) is the base of the triangle.

3. **Use the Tangent Function**:
The tangent of the angle of elevation (θ) can be defined as the ratio of the opposite side (height of the flagstaff) to the adjacent side (length of the shadow):
\[ \tan(θ) = \frac{\text{Height}}{\text{Shadow Length}} = \frac{6}{2\sqrt{3}} \]

4. **Simplify the Expression**:
Now, simplify the fraction:
\[ \tan(θ) = \frac{6}{2\sqrt{3}} = \frac{6}{2} \cdot \frac{1}{\sqrt{3}} = 3 \cdot \frac{1}{\sqrt{3}} = \frac{3}{\sqrt{3}} \]
We can further simplify this:
\[ \tan(θ) = \frac{3}{\sqrt{3}} = \sqrt{3} \]

5. **Find the Angle θ**:
Now, we need to find the angle whose tangent is \(\sqrt{3}\). From trigonometric values, we know:
\[ \tan(60^\circ) = \sqrt{3} \]
Therefore,
\[ θ = 60^\circ \]

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