An airplane flies at an altitude of y = 5 miles toward a point directly over an observer. The speed of the plane is 500 miles per hour. Find the rates at which the angle of elevation θ is changing when the angle is θ = 45°, θ = 60°, and θ = 80°.

Answer 1

MathMate had a very nice solution to this question : )
Let
angle of elevation=θ
height of plane = H
horizontal distance from observer = x
tan(θ)=x/H

Use implicit differentiation
d(tan(θ))/dt = d(x/H)/dt
sec²(θ)dθ/dt = (dx/dt)/H
dθ/dt=(1/(Hsec²(θ))(dx/dt)
dθ/dt=(cos²(θ)/H)*(dx/dt)