What is the distance between the plane and the base of the arch? whats the distance between a point on the ground directly below the plane and the base of arch?

Question

A plane is flying within sight of the Gateway Arch in St. Louis, Missouri, at an elevation of 35,000 ft. The pilot would like to estimate her distance from the Gateway Arch. She finds that the angle of depression to a point on the ground below the arch is 25°. (Round your answers to the nearest foot.)

Answer 1

We can set up a right triangle to solve this problem. The triangle's vertices will be the plane, the base of the arch, and the point in the air above the arch that is level with the plane. The angle of depression measures how far you'd have to look downwards from the plane to see the base of the arch. So, the measure of the vertex located at the plane is the same as the angle of depression: 25°. The length of the leg from the base of the arch to the point in the air above the arch is 35,000 feet, since it's level with the plane.

 

 

For the first question, we'd like to find the distance between the plane and the base of the arch. That's the hypotenuse of the triangle. So, we should use sine:

 

sine = opposite/hypotenuse

sin(25°) = 35000/x

0.423 = 35000/x

0.423x = 35000

x = 82817.06

 

So, the distance between the plane and the base of the arch is 82,817 feet.

 

 

For the second question, we'd like to find the distance between a point on the ground directly below the plane and the base of the arch. That's equal to the other leg of the triangle (the distance from the plane to a point in the air above the arch). So, we should use tangent:

 

tangent = opposite/adjacent

tan(25°) = 35000/x

0.466 = 35000/x

0.466x = 35000

x = 75057.74

 

So, the distance between a point on the ground directly below the plane and the base of the arch is 75,058 feet.