If the elevation of the Sun changes from `30^(@)` to `60^(@)` , then the difference between the lengths of shadows of a pole 15 metre high is

Options

A. 7. 5 metre
B. 15 metre
C. `10sqrt(3)` metre
D. `5sqrt(3)` metre

Answer

To find the difference between the lengths of the shadows of a pole 15 meters high when the angle of elevation of the Sun changes from \(30^\circ\) to \(60^\circ\), we can follow these steps:

### Step-by-Step Solution

1. **Identify the problem**: We have a pole of height \(AB = 15\) meters. We need to find the lengths of the shadows when the angle of elevation of the Sun is \(30^\circ\) and \(60^\circ\).

2. **Set up the triangles**:
- For the angle of elevation \(30^\circ\), let the length of the shadow be \(BD\).
- For the angle of elevation \(60^\circ\), let the length of the shadow be \(BC\).

3. **Use the tangent function**:
- For triangle \(ABD\) (where \(A\) is the top of the pole, \(B\) is the base of the pole, and \(D\) is the tip of the shadow when the angle is \(30^\circ\)):
\[ \tan(30^\circ) = \frac{AB}{BD} \]
Substituting the known values:
\[ \tan(30^\circ) = \frac{15}{BD} \]
Since \(\tan(30^\circ) = \frac{1}{\sqrt{3}}\):
\[ \frac{1}{\sqrt{3}} = \frac{15}{BD} \]
Cross-multiplying gives:
\[ BD = 15\sqrt{3} \text{ meters} \]

4. **Calculate the shadow length for \(60^\circ\)**:
- For triangle \(ABC\) (where \(C\) is the tip of the shadow when the angle is \(60^\circ\)):
\[ \tan(60^\circ) = \frac{AB}{BC} \]
Substituting the known values:
\[ \tan(60^\circ) = \frac{15}{BC} \]
Since \(\tan(60^\circ) = \sqrt{3}\):
\[ \sqrt{3} = \frac{15}{BC} \]
Cross-multiplying gives:
\[ BC = \frac{15}{\sqrt{3}} \text{ meters} \]

5. **Find the difference between the lengths of the shadows**:
- The difference \(CD\) between the lengths of the shadows is:
\[ CD = BD - BC \]
Substituting the values we found:
\[ CD = 15\sqrt{3} - \frac{15}{\sqrt{3}} \]
To combine these, we need a common denominator:
\[ CD = 15\sqrt{3} - \frac{15}{\sqrt{3}} = 15\sqrt{3} - \frac{15\sqrt{3}}{3} = 15\sqrt{3} - 5\sqrt{3} = 10\sqrt{3} \text{ meters} \]

### Final Answer
The difference between the lengths of the shadows of the pole is \(10\sqrt{3}\) meters.

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