Satellite question

Question

An earth-orbiting satellite is passing between the oak ride laboratory in Tennessee and the Langley research center in Virginia, which are 446 miles apart. If the angles of elevation to the satellite from the oak ride and Langley facilities are 58° and 72°, respectively, how far is the satellite from each station?

Answer 1

angles 58 and 72 degrees

distance = 446 miles

solve the ASA triangle for the other two sides

ASA means Angle-Side-Angle where Side is the side between the two angles

side opposite angle 58 is 493.7 miles

553.7 miles is opposite angle 72

use a triangle calculator on line

or use the Law of sines or cosines

c^2 = a^2 +b^2 =2abCosC

requires knowing 2 sides

Law of Sines requires knowing 2 angles

3 triangle angles sum to 180 degrees

3rd angle = 180-72-58= 50 degrees = angle opposite 446 miles

446/sin50 = a/sin58

a = 446sin58/sin50=about 493.74348 = about 493.7 miles

b = 446sin72/sin50 =about 553.7161846 = about 553.7

these calculations ignore the earth's curvature

or assumes the 446 miles is the straight line distance, through the earth between the two facillities

although using the curvature distance doesn't change the results that much