Question

Two geological field teams are working in a remote area. A global positioning system (GPS) tracker at their base camp shows the location of the first team as 39 km away, 13° north of west, and the second team as 31 km away, 36° east of north. When the first team uses its GPS to check the position of the second team, what does the GPS give for the following?

(a) the second team's distance from the first team____ km

(b) the second team's direction from the first team, measured relative to due east
_______° east of due north, east of due south, south of due east, or north of due east

Answer 1

use the law of cosines to determine the distance

Distance from base camp to first team = a = 39 km and angle = 13 deg North of west
Distance from base camp to second team = b = 31 km and angle 36 deg east of north
Angle at base camp between the teams = C = 90 - 13 + 36 =113 deg
Consider the triangle formed by the points base camp, team A and Team B. Then the distance between the teams will be,
c^2 = a^2 + b^2 - 2*a*b*Cos(C)
c^2 = 39^2 + 31^2 - (2 * 39 * 31 * Cos(113)
c^2 = 1521 + 961 - 2418*(-0.3907)
c^2 = 3426.8
c = 58.5387 km

(b) law of sines

sin(C)/c = sin(B)/b
sin(B) = (b/c)sin(C) = (31/58.5387) * 0.89879 = 0.48746
B = sin-1 (0.48746)
B = 29.174 degrees

So the angle will be 27.91752 deg north of due east from camp 1.