Rate of change of distance: plane w/ constant speed 19km/min passes 10km over radar,

sktsasus

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"A plane flying with a constant speed of 19km/min passes over a ground radar station at an altitude of 10km and climbs at an angle of 20 degrees. At what rate is the distance from the plane to the radar station increasing 2 minutes later?"


So I drew up a triangle with a vertical height 10km and an angle of elevation of 20 degrees. But I'm not sure how to proceed after this. What equation do I have to set up so that I can implicitly differentiate it? How would I relate the triangle into it?


Any help?
 
You must remember your geometry and trigonometry.

Given 2 sides of the triangle, which you have, and an appropriate angle, which you have, find the length of the third side.

We haven't even done any calculus, yet. You just have to get that first part. Let's see it.
 
I always begin this type of exercise by introducing a coordinate system.

We can let the horizontal axis represent elapsed time (i.e., we're re-naming the x-axis and now calling it the t-axis). Points in this system have (t, y) coordinates.

Put the radar station at the Origin (0, 0).

At time zero, the plane is at (0, 10).

Now, you need to find an expression (in terms of t) for the y-coordinate of the plane's position at other times. That's trigonometry.

Once you have an expression for the plane's altitude at time t, use the Pythagorean Theorem, to find an expression for the distance between the plane and the radar station at time t.

That expression gives a function for the distance, and you will find it's derivative, to get the rate of change. :cool:
 
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