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Question
An airplane flies at an altitude of 5 miles toward a point directly over an observer. The speed of the plane is 600 miles/hour. Fine the rates at which the angle of elevation is changing when the angle = 30 degrees.
The airplane is 5 miles straight up.
The speed of the airplane is 600 m/h.
A little later on, the angle of elevation of the plane is 30 degrees,
so the plane is no longer directly overhead.
Let x be the horizontal distance, y be the vertical distance. and let θ be the angle.
If we take a triangle with x along the ground, θ is the angle of elevation.
It is known that tan(θ) = y/x and that y does not vary.
Taking the derivative of both sides yields sec²(θ)dθ = -y/x².
It is known that the hypoteneuse of this triangle has length √(x²+y²).
Using this, it is known that sec(θ) = √(x²+y²) / x.
Using this, it can be seen that sec²(θ) = (x²+y²)/x².
Putting this in gives (x²+y²)/x² dθ = -y/x².
Note that both sides are divided by x², so that can be cancelled.
This leaves (x²+y²)dθ = -y, so dθ = -y/(x²+y²) where x = 600√3 and y = 600.
That is the change in angle.
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