If a flag-staff of 6 m height placed on the top of a tower throws a shadow of `2sqrt(3)` m along the ground, then what is the angle that the sun makes with the ground?

Options

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

Answer

To solve the problem, we need to determine the angle that the sun makes with the ground based on the height of the flagstaff and the length of the shadow it casts. Here’s a step-by-step solution:

### Step 1: Understand the Problem
We have a flagstaff of height 6 m on top of a tower, and it casts a shadow of length \(2\sqrt{3}\) m. We need to find the angle \(\theta\) that the sun makes with the ground.

### Step 2: Draw the Diagram
Draw a right triangle where:
- The vertical side (perpendicular) represents the height of the flagstaff (6 m).
- The horizontal side (base) represents the length of the shadow (\(2\sqrt{3}\) m).
- The angle \(\theta\) is the angle between the ground and the line from the top of the flagstaff to the tip of the shadow.

### Step 3: Use the Tangent Function
In a right triangle, the tangent of an angle is given by the ratio of the opposite side (height of the flagstaff) to the adjacent side (length of the shadow):
\[ \tan(\theta) = \frac{\text{Opposite}}{\text{Adjacent}} = \frac{6}{2\sqrt{3}} \]

### Step 4: Simplify the Expression
Now, simplify the right-hand side:
\[ \tan(\theta) = \frac{6}{2\sqrt{3}} = \frac{6}{2} \cdot \frac{1}{\sqrt{3}} = 3 \cdot \frac{1}{\sqrt{3}} = \frac{3}{\sqrt{3}} = \sqrt{3} \]

### Step 5: Find the Angle
We know that:
\[ \tan(60^\circ) = \sqrt{3} \]
Thus, we can conclude:
\[ \theta = 60^\circ \]

### Final Answer
The angle that the sun makes with the ground is \(60^\circ\).

---