To solve this problem, we need to determine the horizontal distance ([tex]\( x \)[/tex]) that the airliner should begin its descent from, given the altitude and the angle of descent. This involves using trigonometry, specifically the tangent function.
### Step-by-Step Solution:
1. Identify the trigonometric relationship:
- In a right triangle, the tangent of an angle is the ratio of the length of the opposite side to the length of the adjacent side.
- Here, the angle of descent (2.5°) is given, the altitude (29,000 feet) is the opposite side, and the horizontal distance ([tex]\( x \)[/tex]) is the adjacent side. Therefore, we can write the relationship as:
[tex]\[
\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}
\][/tex]
where [tex]\( \theta = 2.5° \)[/tex], the opposite side is 29,000 feet, and the adjacent side is [tex]\( x \)[/tex].
2. Set up the equation:
[tex]\[
\tan(2.5°) = \frac{29000}{x}
\][/tex]
3. Solve for [tex]\( x \)[/tex]:
[tex]\[
x = \frac{29000}{\tan(2.5°)}
\][/tex]
4. Calculate [tex]\( \tan(2.5°) \)[/tex]:
- Using a calculator to find the tangent of 2.5 degrees:
[tex]\[
\tan(2.5°) \approx 0.04366
\][/tex]
5. Substitute this value into the equation:
[tex]\[
x = \frac{29000}{0.04366}
\][/tex]
6. Perform the division:
[tex]\[
x \approx \frac{29000}{0.04366} \approx 664,512 \text{ feet}
\][/tex]
### Conclusion:
The horizontal distance from the runway at which the airliner should begin its descent is approximately 664,512 feet.
