Problema Solution
two aircraft are flying within the vicinity of a control tower. The first aircraft is 15 km away from the tower at a bearing of 25° and heading due west. The second aircraft is 12 km away from the tower at a bearing of 200° and heading north-east. Determine the position (bearing and distance) relative to the tower that the paths of these two aircraft will cross.
Hint: Find linear functions that represent the paths of the two aircraft.
Answer provided by our tutors
the linear function of the first plane
the plane goes trough the point (0, 15*cos(25)) and it is parallel to x-axis
y = 15*cos(25)
y = 13.59
the linear function of the second plane
the plane goes trough the point (-12*cos(70), -12*sin(70)) and has a slope m = tan(45) = 1 (since it is heading north-east)
y = x + b
-12*sin(70) = -12*cos(70) + b
b = 12(cos(70) - sin(70))
b = -7.17
y = x - 7.17
by solving the system of equations
y = 13.59
y = x - 7.17
we find
x = 20.76
y = 13.59
using the Pythagorean Theorem we find the distance from the tower
d = (20.76^2 + 13.59^2)^0.5 => d = 24.81 km
the bearing is arctan(20.76/13.59) = arctan(1.5275) = 56.79 degrees.