P is the plane's position
R is the radar station's position
V is the point located vertically of the radar station at the plane's height
h is the plane's height
d is the distance between the plane and the radar station
x is the distance between the plane and the V point
Since the plane flies horizontally, we can conclude that PVR is a right triangle. Therefore, the pythagorean theorem allows us to know that d is calculated:
d=(h^2+x^2)^(1/2)
We are interested in the situation when d=2mi, and, since the plane flies horizontally, we know that h=1mi regardless of the situation.
We are looking for
(dd/dt)=d'.
d^2=h^2+x^2
d'=(x*x')/d
We can calculate that, when d=3mi
x=(d^2−h^2)^(1/2)=(3^2−2^2)^(1/2)=(5)^(1/2) m
Knowing that the plane flies at a constant speed of 510mi/h, we can calculate:
d'=(x*x')/d=(((5)^(1/2))*(510))/3=
380.13 mi/h.
the rate at which the distance from the plane to the station is increasing when it is 3 mi away from the station is 380 mi/h
