Question

an airplane is flying at an altitude of 8,000 feet when the pilot sees the runway ahead at an angle of depression of 20°. to the nearest foot, what is the horizontal distance to the runway?
the horizontal distance to the runway, to the nearest foot, is \\(\square\\) feet.

Explanation:

Step1: Understand the trigonometric relationship

We can model this situation using a right triangle, where the altitude of the airplane (8000 feet) is the opposite side to the angle of depression (20°), and the horizontal distance to the runway is the adjacent side. The tangent of an angle in a right triangle is given by the ratio of the opposite side to the adjacent side. So, \(\tan(\theta)=\frac{\text{opposite}}{\text{adjacent}}\). Here, \(\theta = 20^{\circ}\), opposite side \(= 8000\) feet, and adjacent side \(= x\) (the horizontal distance we need to find). So, \(\tan(20^{\circ})=\frac{8000}{x}\).

Step2: Solve for \(x\)

We can rearrange the formula to solve for \(x\). From \(\tan(20^{\circ})=\frac{8000}{x}\), we get \(x=\frac{8000}{\tan(20^{\circ})}\). We know that \(\tan(20^{\circ})\approx0.3640\) (using a calculator). So, \(x = \frac{8000}{0.3640}\approx21978\) (rounded to the nearest foot).

Answer:

21978