A kite 100 ft above the ground moves horizontally at a speed of 10 ft/s. At what rate is the angle (in radians) between the string and the horizontal decreasing when 200 ft of string have been let out?

Answer 1

if the angle is θ, and the string length is s, and the horizontal distance the kite has flown is x,

sinθ = 100/s
cosθ dθ/dt = -100/s^2 ds/dt

x^2 + 100^2 = s^2
2x dx/dt = 2s ds/dt, so
ds/dt = x/s dx/dt = cosθ dx/dt

so, dθ/dt = -100/s^2 dx/dt

so, when s=200,
dθ/dt = -100/400^2 * 10 = -1/40 rad/s