A plane flying horizontally at an altitude of 2 miles and a speed of 520 mi/h passes directly over a radar station. Find the rate at which the distance from the plane to the station is increasing when it has a total distance of 5 miles away from the station. (Round your answer to the nearest whole number.)

To solve this problem, we need to find the rate at which the distance from the plane to the radar station is increasing at a specific moment. This involves related rates and the Pythagorean theorem.

Let's set up the problem:

1. **Variables:**
   - \( x \): The horizontal distance between the plane and the radar station (in miles).
   - \( z \): The distance from the plane to the radar station (in miles).
   - \( h \): The altitude of the plane, which is 2 miles.
   - \( \frac{dx}{dt} \): The horizontal speed of the plane, which is 520 mi/h.
   - \( \frac{dz}{dt} \): The rate at which the distance from the plane to the radar station is increasing, which we need to find.

2. **Relationships:**
   - Using the Pythagorean theorem, we have:
     \[
     z^2 = x^2 + h^2
     \]
   - Since the altitude \( h \) is constant (2 miles), we can differentiate both sides of the equation with respect to time \( t \):
     \[
     \frac{d}{dt}(z^2) = \frac{d}{dt}(x^2 + h^2)
     \]
     This simplifies to:
     \[
     2z \frac{dz}{dt} = 2x \frac{dx}{dt}
     \]

3. **Simplify and solve for \( \frac{dz}{dt} \):**
   \[
   z \frac{dz}{dt} = x \frac{dx}{dt}
   \]
   \[
   \frac{dz}{dt} = \frac{x}{z} \cdot \frac{dx}{dt}
   \]

4. **Values at the given moment:**
   - When the total distance \( z \) is 5 miles.
   - The altitude \( h \) is 2 miles.
   - The horizontal distance \( x \) can be found using the Pythagorean theorem:
     \[
     z^2 = x^2 + h^2
     \]
     \[
     5^2 = x^2 + 2^2
     \]
     \[
     25 = x^2 + 4
     \]
     \[
     x^2 = 21
     \]
     \[
     x = \sqrt{21}
     \]

5. **Substitute the values:**
   \[
   \frac{dz}{dt} = \frac{\sqrt{21}}{5} \cdot 520
   \]

6. **Calculate \( \frac{dz}{dt} \):**
   \[
   \frac{dz}{dt} = \frac{\sqrt{21}}{5} \cdot 520 \approx 478.94
   \]

Rounding to the nearest whole number, the rate at which the distance from the plane to the radar station is increasing when it is 5 miles away is approximately:

\[
\boxed{479 \text{ mi/h}}
\]

Would you like further details or have any questions?

Here are 8 related questions to expand on this information:
1. How does changing the altitude of the plane affect the rate at which the distance to the radar station increases?
2. What if the speed of the plane changes, how would that impact the result?
3. How can you calculate the rate of change of the angle of elevation of the plane from the radar station?
4. How would the problem change if the radar station were moving as well?
5.

Explore how to calculate the rate at which the distance from a plane to a radar station changes using related rates and the Pythagorean theorem. Get a detailed solution with step-by-step calculations. Suitable for Grades 11-12.

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