How fast does the angle of elevation of your head change when the horizontal distance between you and the bird is 80 m?

Question

You are stationary on the ground and are watching a bird fly horizontally towards you at a rate of 3 m/s. The bird is located 30 m above your head. How fast does the angle of elevation of your head change when the horizontal distance between you and the bird is 80 m? (Leave your answer as an exact number.)

I keep getting this wrong, please help me answer with a detailed explanation.

Thank you,

Answer 1

Let x (t) is horizontal distance from you to beard and θ is angle of elevation of your head. Then tanθ = 30/x;

We differentiate now: sec²θdθ/dt = -30/x²dx/dt; dθ/dt = - 30/x²dx/dt·cos²θ.

dx/dt = - 3m/s, x = 80 m; cos²θ = 80²/(80² + 30²).

So, dθ/dt = -30/80²·(- 3)(80²/(80² + 30²) = 90/(80² + 30²) = 9/730 rad/s