A kite 100 ft above the ground moves horizontally at a speed of 12 ft/s. At what rate is the angle (in radians) between the string and the horizontal decreasing when 200 ft of st

Question

A kite 100 ft above the ground moves horizontally at a speed of 12 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?

Please show steps. I have been stuck on this problem forever.

Answer 1

Let Θ be the angle between the string and the horizontal. dx/dt =12 ft /sec length of string = 200 ft and the height of the kite is 100 ft. Using these numbers let tanΘ = 100/x where x is the length of the horizontal. Take the derivative of tanΘ which gives sec²ΘdΘ/dt = -100 dx/dt/ x² now secΘ =200/x and sec²Θ = 4(10)⁴/x². Going back to the previous equation and solving for dΘ/dt = -100 * 12 x²/ 4(10)⁴x² the x² cancel leaving

dΘ/dt = - 1200/ 40000 rad/sec and this gives dΘ/dt = -.03 rad/sec