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To solve this problem, we can break it down into a few steps by setting up a coordinate system and using the concept of related rates.

1. **Define Position and Variables**: 
   - Let the airport be at the origin (0, 0).
   - The first airliner passes over the airport at noon, traveling east at 530 mph.
   - The second airliner passes over the airport at 1:00 p.m., traveling south at 570 mph.
   - We want to find the rate at which the distance between them is changing at 3:00 p.m.

2. **Determine Positions of Each Airliner at 3:00 p.m.**:
   - The first airliner has been flying for 3 hours (since it passed at noon).
   - The second airliner has been flying for 2 hours (since it passed at 1:00 p.m.).

   Let:
   - \( x(t) \) be the position of the first airliner along the eastward (x-axis) at time \( t \).
   - \( y(t) \) be the position of the second airliner along the southward (y-axis) at time \( t \).

   At 3:00 p.m.:
   - The first airliner’s position is \( x = 530 \times 3 = 1590 \) miles east.
   - The second airliner’s position is \( y = 570 \times 2 = 1140 \) miles south.

3. **Calculate the Distance \( D \) between the Airliners**:
   - The distance \( D \) between the two airliners at 3:00 p.m. is given by:
     \[
     D = \sqrt{x^2 + y^2} = \sqrt{1590^2 + 1140^2}
     \]

4. **Differentiate to Find \( \frac{dD}{dt} \)**:
   - Using the Pythagorean theorem, we have \( D = \sqrt{x^2 + y^2} \).
   - Differentiating both sides with respect to \( t \), we get:
     \[
     \frac{dD}{dt} = \frac{1}{2\sqrt{x^2 + y^2}} \cdot 2(x \frac{dx}{dt} + y \frac{dy}{dt})
     \]
   - Since \( \frac{dx}{dt} = 530 \) mph (east) and \( \frac{dy}{dt} = 570 \) mph (south), substitute these values along with \( x = 1590 \) miles and \( y = 1140 \) miles to find \( \frac{dD}{dt} \).

Let’s calculate each part of this setup.The distance \( D \) between the two airliners at 3:00 p.m. is approximately 1956.45 miles. The rate at which the distance between them is changing at that time, \( \frac{dD}{dt} \), is approximately 762.86 miles per hour.

Would you like more details on any part of this solution?

Here are five related questions that expand on this topic:
1. How would the solution change if one of the planes was flying at a different altitude?
2. How would you solve this problem if the airliners were moving at non-constant speeds?
3. What if the second plane passed over the airport at a different time, say 2:00 p.m.?
4. How could you adapt this solution if the two planes were traveling in directions that weren’t exactly east and south?
5. How does the concept of related rates help in solving other real-world problems involving changing distances?

**Tip**: In related rates problems, it's essential to carefully define each variable and differentiate using the chain rule, keeping units consistent for accurate results.

An airliner passes over an airport at noon traveling 530 mi/hr due east. At 1:00 p.m., another airliner passes over the same airport at the same elevation traveling due south at 570 mi/hr. Assuming both airliners maintain their (equal) elevations, how fast is the distance between them changing at 3:00 p.m.?

Learn how to calculate the rate of change of distance between two airliners moving in perpendicular directions. This problem involves related rates, the Pythagorean theorem, and differentiation techniques. Suitable for students studying advanced high school or college-level calculus.

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