Akite 100 ft above the ground moves horizontally at a speed of 8 ft/s. at what rate is the angle between the string and the horizontal decreasing when 200 ft of string has been let out

After you've diagrammed the scenario, you can see that a right triangle forms after 200 ft of string had been let out. Now let the angle formed between the string and the horizontal plane be θ (now, the hypotenuse = 200 ft, and with reference to the θ, opposite side = 100 ft) so we have, sin θ = opp / hyp = 100/200 = 1/2 therefore, θ = 30 now horizontal distance travelled by the kite (again assuming the person flying the kite is still/not moving) or adjacent side = cos θ x hypotenuse = (cos 30) x 200 ft = 173 ft time elapsed so far = distance / speed = 173 ft / (8 ft/s) = 21.66 secs rate at which angle is decreasing = total angle covered / time elapsed = 30/21.66 = 1.39 degrees / sec you may want to convert it to radians if that's what you are looking for. 1.39 degrees/sec = 1.39 x ( π / 180) = 0.024 radians/sec