A Global Positioning System satellite orbits a = 13,000 miles above Earth's surface (see figure).Find the angle of depression from the satellite to the horizon.Assume the radius of Earth is 4000 miles.Round your answer to two decimal places.
A)77.39º
B)78.39º
C)79.39º
D)76.39º
E)80.39º

Question

A Global Positioning System satellite orbits a = 13,000 miles above Earth's surface (see figure).Find the angle of depression from the satellite to the horizon.Assume the radius of Earth is 4000 miles.Round your answer to two decimal places. ​  ​
A)77.39º 
B)78.39º 
C)79.39º 
D)76.39º 
E)80.39º

Quizplus · Accepted Answer

Let O be the center of the Earth and S be the location of the satellite, as shown in the figure. Let H be the point where the line of sight from the satellite to the horizon intersects the surface of the Earth. Then, OH = 4000 miles (radius of the Earth) and OS = 13000 miles (altitude of the satellite).

[asy]
size(150);
import TrigMacros;
import olympiad;
real R=1;
pair O=(0,0), S=(R,0), H=(R*sqrt(3),0);
draw(circle(O,R));
draw(O--S--H--O--(H.x,0)--cycle);
draw(rightanglemark(S,H,(H.x,0)));
label("$O$",O,N);
label("$S$",S,NE);
label("$H$",H,N);
label("$a$",S--O,SW);
[/asy]

The angle of depression is the angle between the line of sight from the satellite to the horizon and the horizontal plane (tangent to the surface of the Earth at H). This angle is equal to the angle between OS and the tangent plane at H. Let A be the point of tangency, and let M be the midpoint of OH. Then, triangle OMA is a right triangle with angles at O and M equal to 90º, and angle AOM equal to the angle of depression.

[asy]
size(150);
import TrigMacros;
import olympiad;
pair O=(0,0), S=(1.3,0), H=(1,0);
pair A=foot(H,O,S);
pair M=(O+H)/2;
draw(circle(O,1));
draw(O--S--H--cycle);
draw(rightanglemark(S,H,A,1));
draw(rightanglemark(O,M,A,1));
label("$O$",O,NW);
label("$S$",S,NE);
label("$H$",H,SE);
label("$A$",A,N);
label("$M$",M,NE);
label("$a$",S--O,SW);
label("$4000$",(O+M)/2,NW);
[/asy]

We have AM = 4000 miles (radius of the Earth), OM = OH/2 = 6000 miles, and OA = AS - OS = R - a = 3000 miles. Therefore, by the Pythagorean Theorem in triangle OMA, we have:

$$\cos AOM = \frac{OA}{OM} = \frac{1}{2}$$

Hence, the angle of depression is:

$$AOM = \boxed{76.39º}$$

A Global Positioning System Satellite Orbits a = 13,000 Miles