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The problem describes the motion of a kite flying 100 feet above the ground, moving horizontally at a speed of 2 ft/s. It asks for the rate at which the angle between the string and the horizontal is decreasing when 200 feet of string has been let out.

Let's define the variables:
- Let \(\theta\) be the angle between the string and the horizontal (in radians).
- The height of the kite, \(y = 100\) ft (constant).
- The length of the string, \(s = 200\) ft (when specified).
- The horizontal distance between the kite and the person holding the string is \(x\).

We have the right triangle formed by the string, the horizontal distance, and the height of the kite. By the Pythagorean theorem:
\[
s^2 = x^2 + y^2
\]
Given \(y = 100\) ft, we can differentiate with respect to time to find \(\frac{d\theta}{dt}\).

I'll calculate this next.The rate at which the angle between the string and the horizontal is decreasing is \(\frac{1}{50}\) radians per second (or 0.02 rad/s).

Would you like further details on this solution or have any questions?

Here are some related questions for further exploration:
1. How does the rate change if the kite moves faster?
2. What happens to the angle as more string is let out?
3. How would the rate of change be affected if the kite's height were to change?
4. Can the rate be negative, and what would that mean in this context?
5. How would the solution differ if the kite’s string length were changing?

**Tip:** Understanding implicit differentiation is key in related rates problems like this one!

A kite 100 ft above the ground moves horizontally at a speed of 2 ft/s. At what rate (in rad/s) is the angle (in radians) between the string and the horizontal decreasing when 200 ft of string have been let out?

Learn how to find the rate of change of the angle between a kite string and the horizontal when a kite is 100 ft above the ground, moving horizontally at 2 ft/s, with 200 ft of string let out. This problem involves related rates, implicit differentiation, and trigonometry.

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