When angle of elevation of the Sun increases from `30^(@)` to `60^(@)`, Shadow of a pole is diminised by 5 meter. The height of the pole is

Options

A. `(5sqrt(3))/2` m
B. `(2sqrt(3))/5` m
C. `2/(5sqrt(3))` m
D. `4/(5sqrt(3))` m

Answer

To solve the problem, we need to find the height of the pole based on the given angles of elevation and the change in the length of the shadow. Let's break it down step by step.

### Step 1: Understanding the Problem
We have a pole whose height we need to find. The angle of elevation of the sun changes from \(30^\circ\) to \(60^\circ\), and during this change, the length of the shadow decreases by 5 meters.

### Step 2: Setting Up the Triangles
1. **For \(30^\circ\):**
- Let the height of the pole be \(h\).
- The length of the shadow when the angle of elevation is \(30^\circ\) can be found using the tangent function:
\[ \tan(30^\circ) = \frac{h}{\text{shadow length at } 30^\circ} \]
- We know that \(\tan(30^\circ) = \frac{1}{\sqrt{3}}\), so:
\[ \frac{h}{\text{shadow length at } 30^\circ} = \frac{1}{\sqrt{3}} \implies \text{shadow length at } 30^\circ = h \cdot \sqrt{3} \]

2. **For \(60^\circ\):**
- Similarly, for \(60^\circ\):
\[ \tan(60^\circ) = \frac{h}{\text{shadow length at } 60^\circ} \]
- We know that \(\tan(60^\circ) = \sqrt{3}\), so:
\[ \frac{h}{\text{shadow length at } 60^\circ} = \sqrt{3} \implies \text{shadow length at } 60^\circ = \frac{h}{\sqrt{3}} \]

### Step 3: Relating the Shadows
According to the problem, the difference in shadow lengths is 5 meters:
\[ \text{shadow length at } 30^\circ - \text{shadow length at } 60^\circ = 5 \]
Substituting the expressions we found:
\[ h \cdot \sqrt{3} - \frac{h}{\sqrt{3}} = 5 \]

### Step 4: Simplifying the Equation
To simplify, we can find a common denominator:
\[ \frac{h \cdot 3}{\sqrt{3}} - \frac{h}{\sqrt{3}} = 5 \]
This simplifies to:
\[ \frac{3h - h}{\sqrt{3}} = 5 \implies \frac{2h}{\sqrt{3}} = 5 \]

### Step 5: Solving for \(h\)
Now, we can solve for \(h\):
\[ 2h = 5\sqrt{3} \implies h = \frac{5\sqrt{3}}{2} \]

### Conclusion
The height of the pole is:
\[ h = \frac{5\sqrt{3}}{2} \text{ meters} \]