To solve this problem, let's follow these steps:
1. Understand the problem:
- The angle of elevation (from the ground to the plane's flight path) is 6 degrees.
- The plane travels 10 miles horizontally from its initial take-off point.
2. Draw a right triangle:
- The horizontal distance (adjacent side to the angle) is 10 miles.
- We need to find the vertical distance (height of the plane, which is the opposite side to the angle).
3. Choose the appropriate trigonometric function:
- Given angle [tex]\(\theta\)[/tex], opposite side (height), and adjacent side (horizontal distance), we use the tangent function:
[tex]\[
\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}
\][/tex]
4. Set up the equation:
- [tex]\(\theta = 6\)[/tex] degrees
- [tex]\(\text{adjacent} = 10\)[/tex] miles
- [tex]\(\tan(6^\circ) = \frac{\text{height}}{10}\)[/tex]
5. Solve for the height:
- Rearrange the equation to solve for height:
[tex]\[
\text{height} = 10 \times \tan(6^\circ)
\][/tex]
6. Convert angle from degrees to radians:
- Since trigonometric functions typically use radians, convert [tex]\(6\)[/tex] degrees to radians:
[tex]\[
6^\circ = 0.1047 \text{ radians} \quad (\text{approximately})
\][/tex]
7. Calculate height using the tangent function:
- [tex]\(\tan(0.1047) \approx 0.1051\)[/tex]
- So,
[tex]\[
\text{height} = 10 \times 0.1051 \approx 1.051
\][/tex]
8. Round to the nearest tenth:
- [tex]\(1.051\)[/tex] rounded to the nearest tenth is [tex]\(1.1\)[/tex].
Therefore, the height of the plane is 1.1.
