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This problem involves related rates, where we have a right triangle formed by the radar antenna, the plane’s horizontal distance from the radar, and the distance between the radar and the plane.

### Given:
- Altitude of the plane (\( y \)) = 6 miles (constant)
- Distance \( s \) from the radar to the plane = 10 miles
- Rate of change of \( s \) (\( \frac{ds}{dt} \)) = 200 miles per hour
- We need to find the horizontal speed (\( \frac{dx}{dt} \)) of the plane.

### Relationship:
We use the Pythagorean theorem to relate \( x \), \( y \), and \( s \):
\[
s^2 = x^2 + y^2
\]

Differentiating both sides with respect to \( t \):
\[
2s \frac{ds}{dt} = 2x \frac{dx}{dt}
\]

Simplifying, we get:
\[
s \frac{ds}{dt} = x \frac{dx}{dt}
\]

### Find \( x \):
From the Pythagorean theorem:
\[
10^2 = x^2 + 6^2
\]
\[
100 = x^2 + 36
\]
\[
x^2 = 64 \implies x = 8 \, \text{miles}
\]

### Find \( \frac{dx}{dt} \):
Substitute the known values into the differentiated equation:
\[
10 \times 200 = 8 \frac{dx}{dt}
\]
\[
2000 = 8 \frac{dx}{dt}
\]
\[
\frac{dx}{dt} = \frac{2000}{8} = 250 \, \text{miles per hour}
\]

### Final Answer:
The speed of the airplane is **250 miles per hour**.

Would you like further details or have any questions?

Here are 5 related questions to deepen your understanding:
1. How would the answer change if the altitude of the airplane were 10 miles instead of 6 miles?
2. What if the rate of change of \( s \) were decreasing instead of increasing?
3. How does the altitude affect the speed of the plane if all other parameters are kept constant?
4. Can you derive a general formula for the airplane's speed based on \( s \), \( y \), and \( \frac{ds}{dt} \)?
5. How would the problem change if the airplane were descending at a constant rate?

**Tip**: In related rates problems, always start by identifying the relationship between variables using geometry or physics and then differentiate to find how the rates are connected.

An airplane flying at an altitude of 6 miles passes directly over a radar antenna. When the airplane is 10 miles away (s = 10), the radar detects that the distance s is changing at a rate of 200 miles per hour. What is the speed (in mi/h) of the airplane?

Learn how to calculate the speed of an airplane flying over a radar antenna using related rates and the Pythagorean theorem. This problem involves applying differentiation to find the airplane's horizontal speed when the distance from the radar is changing. Suitable for high school students in Grades 10-12.

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