The rate at which the angle of elevation is changing

Question

An airplane flies at an altitude of 5 miles toward a point directly over an observer. The speed of the plane is 600mi/h. Find the rate at which the angle of elevation is changing when the angle is 30°

Answer 1

The altitude is 5 miles (y = 5), and the angle of elevation is 30°, so x = 5√3 (cot 30°= √3)

Then, y is a constant, and x = 5√3 - 600t.

Then, θ = cot^-1(x/y) = cot^-1((5√3 - 600t)/5) = cot^-1(√3 - 120t) , so

dθ/dt = 120/(1 + (√3 - 120t)²)

At t=0, this is 120/(1+(√3)²) = 120/4 = 30

Note that this θ measured in radians and time is in hours.

30 radians/hour

Let's say we wanted it in degrees/sec. It would be

30 * 180/π/3600 = 3/(2π) degrees per second