Expert: Paul Klarreich - 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,

You wrote:
Subject:  calculus 2 related rates
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 says that  c^2 = a^2 + b^2 - 2ab cos C]

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On to your example:

NEVER do these without a diagram.  Mine looks like this:

S is the radar station.
A is the first point, directly above S and AS = 1.
B is the present location of the aircraft, to the right of A.
Angle SAB is 120 degrees.


Let x = SB, the distance from the Station to the plane.
   y = AB, the distance from the first point to the plane.

Then dx/dt = 7, and we have to find dy/dt

x and y are variables, and the present value of x is 2.

In these you want a relationship between x and y.  Yes, the L.o.C. looks good here, and it says:

x^2 = y^2 + 1^2 - 2(y)(1) cos 120o

Now cos 120o = -1/2, so that becomes:

x^2 = y^2 + 1 + y

Now we can start the process.  Differentiate implicitly w.r.t. time:

2x dx/dt = 2y dy/dt + dy/dt

2x dx/dt = (2y + 1)dy/dt

Unfortunately, we require a value of y corresponding to x=2, so we have to solve for y in:

2^2 = y^2 + 1 + y
4 = y^2 + 1 + y
y^2 + y + 1 = 4
y^2 + y - 3 = 0

This is a Quadratic Formula case and we get:
   -1 + sqrt(13)
y = -------------  [-sqrt(13) not applicable.]
        2

Now put that, with x=2, and dx/dt = 7 into:

2x dx/dt = (2y + 1)dy/dt

2(2)(7) = (-1 + sqrt(13) + 1) dy/dt

28 = sqrt(13) dy/dt, and:

dy      28      28 sqrt(13)
-- = -------- = -----------
dt   sqrt(13)       13
[if you are expected to rationalize.]