A kite 80 ft above the ground moves horizontally (away from the person holding the string) at a seed of 4 ft/s.

Question

At what rate (in rad/s) is the angle (in radians) between the string and the horizontal changing when 160 ft of string have been let out?

Include an appropriate picture that shows your variables. Give an exact answer with correct units. (Assume one end of the string is on the ground.)

Answer 1

Let s be the length of the string an x be the horizontal distance to the kite.

tan(θ) = 80/x

Take the derivative of both sides

sec²(θ) dθ/dt = -80/x² dx/dt

cos(θ) = x/s

dθ/dt = -80cos²(θ)/x² dx/dt = -80(x²/s²)/x² dx/dt = -(80/s²) dx/dt

when s = 160

dθ/dt = -(80/160²) 4 = -0.0125 radians/s

The negative means the rate is decreasing

Answer 2