To solve this problem, let's follow a step-by-step process:
1. Understand the problem:
- An airplane climbs at an angle of 11° with respect to the ground.
- The airplane reaches an altitude of 400 feet.
- We need to find the ground distance the airplane has traveled, to the nearest foot.
2. Draw a diagram:
- Draw a right triangle where:
- The angle between the hypotenuse (the airplane's path) and the adjacent side (the ground distance) is 11°.
- The opposite side (the altitude) is 400 feet.
- The adjacent side (the ground distance) is unknown.
3. Mathematical Representation:
- In this right triangle, the known values are:
- Angle of elevation [tex]\( \theta = 11° \)[/tex]
- Opposite side (altitude) [tex]\( h = 400 \)[/tex] feet
- We need to find the length of the adjacent side (ground distance) [tex]\( d \)[/tex].
4. Use Trigonometric Relationships:
- The tangent of an angle in a right triangle is defined as the ratio of the opposite side to the adjacent side.
[tex]\[
\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}
\][/tex]
- Plug in the known values:
[tex]\[
\tan(11°) = \frac{400}{d}
\][/tex]
5. Solve for the ground distance [tex]\( d \)[/tex]:
- Rearrange the formula to solve for [tex]\( d \)[/tex]:
[tex]\[
d = \frac{400}{\tan(11°)}
\][/tex]
- Use a calculator to find [tex]\( \tan(11°) \)[/tex]:
[tex]\[
\tan(11°) \approx 0.1944
\][/tex]
- Now, compute the ground distance:
[tex]\[
d = \frac{400}{0.1944} \approx 2057.07 \text{ feet}
\][/tex]
6. Round to the nearest foot:
- The ground distance the airplane has traveled is approximately 2057 feet.
7. Conclusion:
- The airplane has traveled approximately 2057 feet along the ground when it has climbed to an altitude of 400 feet.
This completes the problem-solving process for finding the ground distance an airplane has traveled at an elevation angle of 11° to reach 400 feet in altitude.
