How long is the oil tanker visible on the radar screen of the coast guard?
From a position of 110km northwest of a coast guard station, an oil tanker makes a radio contact with the coast guard. The tanker is travelling at a speed of 25 km/h due south. The radar unit at the coast guard has a range of 90 km. For what length of time can the coast guard expect the tanker to be visible on the radar screen?
Suppose the tanker is heading due south.
When the tanker contacts the coast guard, it is out of range (110 > 90)
We need to find out how far west of the station the tanker is, and how far north it is.
The angle the line between the tanker and the station makes with the east-west line is 45 degrees = pi/4
So the tanker is cos(pi/4) x 110 = 77.78km west of the station and sin(pi/4) x 110 = 77.78km north of the station (it is exactly north-west of the station, so this makes sense)
The tanker comes in to radar range when the line between the tanker and the station is 90km long. It stays the same distance west of the station, as it is heading due south, so it will be
sqrt(90^2 - 77.78^2) = 45.28km north of the station when it comes in to range.
It will stay in range until it is 45.28km south of the station (using symmetry). So it will be in range for
45.28*2 = 90.55km
At a speed of 25km/h this will take 90.55/25 = 3.62 hrs = 3hrs 37 mins