Asked by Anonymous

Hi, I am doing a practice exam, and I came across this problem and I am have difficultly starting it.

A UFO flies horizontally at a constant speed at an altitude of 15 km and passes directly over a tracking telescope on the ground. When the angle of elevation is pi/3, this angle is decreasing at a rate of 0.1 rad/ min. How fast is the UFO flying?

I need help formulating the equation, the hardest part for me. Because I know after I have the equation, I just need to take the derivative implicitly and plug in values into the equation.
Answered by Reiny
Do not attempt any of these types without a diagram.
Draw the ground and the horizontal path of the UFO 15 km above the ground.
Label the tracking station as T, the point directly above T as A, and the position of the UFO as B
(AB will be parallel to the ground, 15 km above it, right ?)

Let AB = x and TB = y
Let Ø be the angle of elevation, the angle ABT = Ø

then x^2 + 15^2 = y^2

tanØ = 15/x or x/15 = cotØ

plug in Ø = pi/3 to find x, then you can find y

you will have to know how to take the derivative of cotØ which is csc^2 Ø, and that is why we also have to know the value of y.
(I got 2 km/min for the speed of the UFO, check that)

Answered by Reiny
In my 3rd last line, if left out a negative sing
d(cotx)/dx = -csc^2 x , as I worked out my 2 km/min, I did use the negative )
Answered by Anonymous
Yes! Thank you very much! I got the exact answer. ;DD!
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