a kite 50 ft above the ground moves horizontally at a speed of 2ft/s. At what rate is the angle between the string and the horizontal decreasing when 200 ft of string has been let out

To solve this problem, we need to use related rates and trigonometry.

### Step 1: Set up the variables
- Let \(\theta\) be the angle between the string and the horizontal ground.
- Let \(s\) be the length of the string, which is given as 200 ft.
- Let \(x\) be the horizontal distance from the point on the ground directly below the kite to the point where the string meets the ground.
- The height of the kite above the ground is constant, given as 50 ft.

From the given information, the kite is moving horizontally at 2 ft/s, which means \(\frac{dx}{dt} = 2 \, \text{ft/s}\).

### Step 2: Apply the Pythagorean theorem
Using the Pythagorean theorem, we have the relationship between the horizontal distance \(x\), the height \(h = 50\) ft, and the string length \(s = 200\) ft:
\[
s^2 = x^2 + h^2
\]
Substituting the known height of the kite:
\[
200^2 = x^2 + 50^2
\]
\[
40000 = x^2 + 2500
\]
\[
x^2 = 37500
\]
\[
x = \sqrt{37500} \approx 193.65 \, \text{ft}
\]

### Step 3: Relating the angle \(\theta\)
The angle \(\theta\) between the string and the horizontal can be related using the trigonometric function:
\[
\cos\theta = \frac{x}{s}
\]
Now differentiate this equation with respect to time \(t\):
\[
\frac{d}{dt} \left( \cos\theta \right) = \frac{d}{dt} \left( \frac{x}{s} \right)
\]
Since the length of the string \(s\) is constant at 200 ft:
\[
-\sin\theta \frac{d\theta}{dt} = \frac{1}{s} \frac{dx}{dt}
\]
Substitute \(\sin\theta = \frac{h}{s} = \frac{50}{200} = \frac{1}{4}\) and \(\frac{dx}{dt} = 2\) ft/s:
\[
-\frac{1}{4} \frac{d\theta}{dt} = \frac{1}{200} \times 2
\]
\[
-\frac{1}{4} \frac{d\theta}{dt} = \frac{1}{100}
\]
Now solve for \(\frac{d\theta}{dt}\):
\[
\frac{d\theta}{dt} = -\frac{4}{100} = -0.04 \, \text{radians per second}
\]

### Final Answer:
The rate at which the angle between the string and the horizontal is decreasing is \(0.04\) radians per second.

Would you like further details or have any questions?

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Here are 5 related questions for deeper exploration:
1. What is the rate of change of the angle if the string length were 300 ft instead of 200 ft?
2. How does the rate of change of the angle depend on the height of the kite?
3. What happens to the rate of change of the angle as the string becomes longer and longer?
4. Can you find the rate at which the string is being let out if the angle decreases by 0.05 radians per second?
5. How would the problem change if the kite were moving vertically rather than horizontally?

**Tip:** When solving related rates problems, always identify which quantities are constant and which ones are changing with time.

A kite 50 ft above the ground moves horizontally at a speed of 2 ft/s. At what rate is the angle between the string and the horizontal decreasing when 200 ft of string has been let out?

Explore how to solve a related rates problem involving a kite 50 ft above the ground moving horizontally. Learn how to calculate the rate at which the angle between the string and the horizontal decreases when 200 ft of string has been let out, using trigonometry and differentiation. Suitable for high school students in Grades 10-12.

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