To determine the angle of elevation of the sun, we'll follow these steps.
1. Understand the Given Information:
- Height of the pole: 7 meters.
- The length of the shadow is [tex]\( \frac{1}{2} \)[/tex] its height, so the shadow length is [tex]\( 2 \times 7 = 14 \)[/tex] meters.
2. Define the Scenario:
- We have a right triangle where:
- The height of the pole is one leg (opposite side of the angle).
- The length of the shadow is the other leg (adjacent side of the angle).
3. Calculate the Angle of Elevation:
- The angle of elevation ([tex]\(\theta\)[/tex]) can be found using the tangent function:
[tex]\[
\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}
\][/tex]
So in our case:
[tex]\[
\tan(\theta) = \frac{7}{14}
\][/tex]
4. Compute the Tangent Value:
- Simplifying,
[tex]\[
\tan(\theta) = \frac{7}{14} = 0.5
\][/tex]
5. Determine the Angle Using Inverse Tangent:
- To find [tex]\(\theta\)[/tex], we use the inverse tangent (arctan) function:
[tex]\[
\theta = \tan^{-1}(0.5)
\][/tex]
6. Convert the Angle from Radians to Degrees:
- When [tex]\(\theta\)[/tex] is calculated, it is typically in radians. To convert it to degrees, we use the conversion ratio [tex]\( \frac{180}{\pi} \)[/tex]:
[tex]\[
\theta \approx 26.56505117707799 \text{ degrees}
\][/tex]
7. Round to the Nearest Degree:
- Finally, we round [tex]\( 26.56505117707799 \)[/tex] to the nearest integer:
[tex]\[
\theta \approx 27 \text{ degrees}
\][/tex]
Thus, the angle of elevation of the sun is approximately [tex]\( 27 \)[/tex] degrees, correct to the nearest degree.
