The shadow of a flagstaff is three times as long as the shadow of the flagstaff when the sun rays meet the ground at `60^(@)` . Find the angle between the sun rays and the ground at the time of longer shadow

Options

A. `45^(@)`
B. ` 30^(@)`
C. ` 15^(@)`
D. ` 90^(@)`

Answer

To solve the problem step by step, we will use trigonometric concepts related to the angles of elevation and the lengths of shadows.

### Step-by-Step Solution:

1. **Understanding the Problem**:
- We have a flagstaff that casts a shadow. When the angle of elevation of the sun is \(60^\circ\), the length of the shadow is \(x\).
- When the angle of elevation changes, the shadow becomes three times longer, which means the new shadow length is \(3x\).
- We need to find the new angle of elevation of the sun, which we will denote as \(\alpha\).

2. **Using Trigonometry for the First Scenario**:
- When the sun's rays make an angle of \(60^\circ\) with the ground, we can use the tangent function:
\[ \tan(60^\circ) = \frac{\text{Height of flagstaff (h)}}{\text{Length of shadow (x)}} \]
- We know that \(\tan(60^\circ) = \sqrt{3}\). Therefore, we can write:
\[ \sqrt{3} = \frac{h}{x} \]
- Rearranging gives us:
\[ h = x \sqrt{3} \]

3. **Using Trigonometry for the Second Scenario**:
- In the second scenario, when the shadow is \(3x\), we can again use the tangent function:
\[ \tan(\alpha) = \frac{h}{3x} \]
- Substituting \(h\) from the previous step:
\[ \tan(\alpha) = \frac{x \sqrt{3}}{3x} = \frac{\sqrt{3}}{3} \]
- We know that \(\tan(30^\circ) = \frac{1}{\sqrt{3}}\), which implies:
\[ \tan(\alpha) = \frac{\sqrt{3}}{3} = \tan(30^\circ) \]

4. **Finding the Angle \(\alpha\)**:
- From the tangent values, we conclude:
\[ \alpha = 30^\circ \]

### Final Answer:
The angle between the sun rays and the ground at the time of the longer shadow is \(30^\circ\).