If a pole of 12 m height casts a shadow of `4sqrt(3)` m long on the ground , then the sun 's angle of elevation at that instant is

Options

A. `30^(@)`
B. `60^(@)`
C. `45^(@)`
D. `90`

Answer

To find the angle of elevation of the sun when a pole of height 12 m casts a shadow of length \(4\sqrt{3}\) m, we can use trigonometric ratios. Specifically, we will use the tangent function, which relates the angle of elevation to the height of the pole and the length of the shadow.

### Step-by-Step Solution:

1. **Identify the given values:**
- Height of the pole (perpendicular) = 12 m
- Length of the shadow (base) = \(4\sqrt{3}\) m

2. **Set up the tangent ratio:**
The tangent of the angle of elevation (\(\theta\)) can be defined as:
\[ \tan(\theta) = \frac{\text{Opposite}}{\text{Adjacent}} = \frac{\text{Height of the pole}}{\text{Length of the shadow}} \]
Substituting the known values:
\[ \tan(\theta) = \frac{12}{4\sqrt{3}} \]

3. **Simplify the expression:**
To simplify \(\frac{12}{4\sqrt{3}}\):
\[ \tan(\theta) = \frac{12 \div 4}{\sqrt{3}} = \frac{3}{\sqrt{3}} = \sqrt{3} \]

4. **Find the angle \(\theta\):**
We know that:
\[ \tan(60^\circ) = \sqrt{3} \]
Therefore, we can conclude:
\[ \theta = 60^\circ \]

5. **Final answer:**
The angle of elevation of the sun at that instant is \(60^\circ\).