Expert: Paul Klarreich - 3/31/2008

Question
An object that is launched vertically is tracked by a radar station located on the ground 4 miles from the launch site. What is the vertical speed of the object at the instant that its distance from the radar station is 6 miles and this distance is increasing at the rate of 6000 miles per hour?

Answer
Questioner:   Ashley
Category:  Calculus
Private:  No
 
Subject:  related rates?
Question:  An object that is launched vertically is tracked by a radar station located on the ground 4 miles from the launch site. What is the vertical speed of the object at the instant that its distance from the radar station is 6 miles and this distance is increasing at the rate of 6000 miles per hour?
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Hi, Ashley,

Is this your first attempt at R-R problems?  If so, the scheme is something like this:

1. Identify the variables in the problem -- the things that change.  Give them names.

2. Write their rates of change as derivatives WITH RESPECT TO time.  Note which are known and which is to be found.

3. Determine a relationship (yes, it is called 'related rates' for a reason) between the variables.  Use a diagram, use your life experience, use your general knowledge and brilliance, do whatever you have to.  This is the key step.

4. Now differentiate implicitly, then substitute the known quantities and rates, and solve for the unknown rate.

AND, Please check the archives for other Related Rates examples.  There are TONS of them.  Click BROWSE PAST ANSWERS.

Variables:

y = vertical distance.
z = distance from station.

rates:  
dy/dt to be found.
dz/dt = 6000 mph

Relation:

4^2 + y^2 = z^2

I think you can handle the rest.  If you have trouble, send me what you did and I'll look at it.