Final answer:
The rate at which the angle between the string and the horizontal is decreasing when 200 ft of string has been let out is 16 ft/s.
Step-by-step explanation:
To find the rate at which the angle between the string and the horizontal is decreasing, we can use trigonometry and related rates. Let's assume that the horizontal distance between the kite and the point where the string is being let out is x, and the length of the string is s. We can set up a right triangle with the string being the hypotenuse, the vertical distance from the kite to the ground being 100 ft, and the horizontal distance from the point where the string is being let out to the ground being x ft.
Using the Pythagorean theorem, we have s^2 = (100 + x)^2 + 8^2. Differentiating both sides with respect to time, we get 2s(ds/dt) = 2(100 + x)(dx/dt), where ds/dt represents the rate at which the string is being let out (which is given as 8 ft/s) and dx/dt represents the rate at which x is changing (which is what we need to find).
Substituting the given values, we have 2(200)(8) = 2(100 + x)(dx/dt). Solving for dx/dt, we can find that the rate at which the angle between the string and the horizontal is decreasing is 16 ft/s when 200 ft of string has been let out.
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