Expert: Ahmed Salami - 2/9/2006

Question
An aircraft climbing at a constant angle of 30o above the
horizontal passes directly over a ground radar station at an altitude of 1 km. At a later instant, the
radar shows that the aircraft is at an air distance of 2 km from the station, and that this distance
is increasing at 7 km/min. What is the speed of the aircraft at that instant? [Note: the law of
cosines c2 = a2 + b2 − 2ab cos  

Answer
Hi Ali,
Sorry for the time it took.
I'm sorry i'm unable to give a sketch of the situation
here, try doing so to get a better understanding of the
solution.
At any time t, let the distance travelled by the
aircraft since passing over the radar station be y and
the air distance from the radar station be x. This
situation can be represented by a triangle with the
lengths 1km, x and y, as well as an angle 120 degrees
opposite to the x length. Making use of this information
and employing the cosine rule to get the length x,
we have
x^2 = 1^2 + y^2 - 2(1)(y)cos120
   = 1^2 + y^2 - 2(1)(y)(-0.5)
   = 1 + y^2 + y
To compare the rate of change of both distances i.e
dx/dt and dy/dt, we simply differentiate both sides with
respect to time t,
Therefore,
2x(dx/dt) = (2y + 1)dy/dt
dy/dt = [2x/(2y + 1)]dx/dt
Now, at the point when the air distance of the aircraft
from the radar station is 2km we require the speed(i.e
the rate of change of the distance travelled by the
aircraft, dy/dt). To do this, we need to find y by
using the equation
x^2 = y^2 + y + 1  (as above)
2^2 = y^2 + y + 1
4 = y^2 + y + 1
y^2 + y - 3 = 0
which can be solved using the quadratic formula to get
y = 1.3 and -2.3
we disregard the negative value because y is a distance
and therefore positive for our purpose.
The aircraft has therefore travelled 1.3km since
passing over the radar station.
But
dx/dt = 7km/min
Using
dy/dt = [2x/(2y + 1)]dx/dt
dy/dt = [2(2)/(2(1.3) + 1)]7
     = [4/(2.6 + 1)]7
     = 28/3.6
     = 7.78km/min
And so we have as our solution, that the speed of the
aircraft at the point when the air distance from the
aircraft is 2km from the radar station is 7.78km/min

I hope i have helped. You can always get back to me.
Regards.