Question

the angle of depression from a descending airplane to the control tower below is 60°. if the plane is 35,000 feet above the ground, approximately what is the planes horizontal distance from the control tower? 70,000 ft. 24,749 ft. 60,622 ft. 20,207 ft.

Explanation:

Step1: Establish the trigonometric relationship

We know that the angle of depression is $60^{\circ}$, and the vertical distance (height of the plane above the ground) is the opposite - side and the horizontal distance from the plane to the control tower is the adjacent - side in a right - triangle. The tangent of an angle in a right - triangle is given by $\tan\theta=\frac{\text{opposite}}{\text{adjacent}}$. Here, $\theta = 60^{\circ}$ and the opposite side $y = 35000$ feet, and we want to find the adjacent side $x$. So, $\tan60^{\circ}=\frac{35000}{x}$.

Step2: Solve for $x$

Since $\tan60^{\circ}=\sqrt{3}\approx1.732$, we can rewrite the equation as $1.732=\frac{35000}{x}$. Cross - multiplying gives us $1.732x = 35000$. Then, $x=\frac{35000}{1.732}\approx20207$ feet.

Answer:

20,207 ft