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Let's solve this problem step-by-step.

### Given:
- Altitude of the plane: \( y = 2 \) miles
- Horizontal speed of the plane: \( \frac{dx}{dt} = 400 \) mi/h
- Distance from the station (hypotenuse): \( s = 4 \) miles

We need to find the rate at which the distance \( s \) from the plane to the station is increasing when \( s = 4 \) miles.

### Step 1: Set Up the Relationship Between Variables
Using the Pythagorean theorem for the right triangle formed by the altitude \( y \), the horizontal distance \( x \) (the distance the plane has traveled horizontally from the station), and the hypotenuse \( s \) (the distance between the plane and the station):

\[
s^2 = x^2 + y^2
\]

### Step 2: Differentiate with Respect to Time \( t \)
We want to find \( \frac{ds}{dt} \). Differentiate both sides of the equation with respect to \( t \):

\[
2s \frac{ds}{dt} = 2x \frac{dx}{dt}
\]

Simplify by dividing both sides by 2:

\[
s \frac{ds}{dt} = x \frac{dx}{dt}
\]

Now, solve for \( \frac{ds}{dt} \):

\[
\frac{ds}{dt} = \frac{x \frac{dx}{dt}}{s}
\]

### Step 3: Substitute Known Values
To use this equation, we need the value of \( x \) when \( s = 4 \) miles.

Using the Pythagorean theorem:

\[
s^2 = x^2 + y^2
\]
\[
4^2 = x^2 + 2^2
\]
\[
16 = x^2 + 4
\]
\[
x^2 = 12
\]
\[
x = \sqrt{12} = 2\sqrt{3} \approx 3.46 \text{ miles}
\]

Now substitute \( x = 2\sqrt{3} \), \( \frac{dx}{dt} = 400 \) mi/h, and \( s = 4 \) miles:

\[
\frac{ds}{dt} = \frac{(2\sqrt{3})(400)}{4}
\]

Simplify:

\[
\frac{ds}{dt} = \frac{800\sqrt{3}}{4}
\]
\[
\frac{ds}{dt} = 200\sqrt{3} \approx 346.41 \text{ mi/h}
\]

### Answer
The rate at which the distance from the plane to the station is increasing is approximately **346.41 mi/h**.

Would you like more details, or do you have any questions?

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Here are five related questions:
1. What would be the rate if the altitude was different?
2. How does the rate of change depend on altitude \( y \)?
3. What is the rate of increase at different values of \( s \)?
4. How would the answer change if the plane's speed was different?
5. How can this setup apply to other real-world tracking problems?

**Tip:** Always use the Pythagorean theorem when working with right-triangle distances in related rates problems!

A plane flying horizontally at an altitude of 2 mi and a speed of 400 mi/h passes directly over a radar station. Find the rate at which the distance from the plane to the station is increasing when the plane is 4 mi from the station.

Learn how to find the rate at which the distance between a plane flying at 400 mi/h at an altitude of 2 mi and a radar station is increasing when the plane is 4 mi from the station. This solution uses the Pythagorean theorem and differentiation techniques, ideal for Grades 11-12.

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