https://file.aiquickdraw.com/mathbot/file/1730431002145pry6jgkg.jpg

To solve this problem, we need to find the rate at which the distance between the two airplanes is changing at 2:30 p.m.

Let's define the setup:

1. **Plane A**: Passes over the airport at noon, traveling west at 520 mi/hr.
2. **Plane B**: Passes over the airport at 1:00 p.m., traveling north at 550 mi/hr.

We want to find the rate at which the distance between the planes is changing at 2:30 p.m.

### Step-by-Step Solution

1. **Define Variables**:
   - Let \( t \) be the time in hours since noon.
   - Let \( x \) represent the westward distance of Plane A from the airport (since it's going west), and \( y \) represent the northward distance of Plane B from the airport.

2. **Express \( x \) and \( y \) in Terms of \( t \)**:
   - For Plane A (starting from noon): \( x = 520t \)
   - For Plane B (starting from 1:00 p.m.): Since Plane B starts an hour later, its travel time is \( t - 1 \), so \( y = 550(t - 1) \) when \( t \geq 1 \).

3. **Distance Between the Two Planes**:
   - Let \( d \) be the distance between the planes. By the Pythagorean theorem:
     \[
     d = \sqrt{x^2 + y^2}
     \]

4. **Differentiate with Respect to \( t \)**:
   - To find the rate at which the distance between the planes is changing, we need to find \( \frac{dd}{dt} \).
   - Differentiating \( d = \sqrt{x^2 + y^2} \) with respect to \( t \):
     \[
     \frac{dd}{dt} = \frac{1}{2\sqrt{x^2 + y^2}} \cdot (2x \cdot \frac{dx}{dt} + 2y \cdot \frac{dy}{dt})
     \]
   - Simplifying, we get:
     \[
     \frac{dd}{dt} = \frac{x \cdot \frac{dx}{dt} + y \cdot \frac{dy}{dt}}{\sqrt{x^2 + y^2}}
     \]

5. **Substitute Values at 2:30 p.m. (t = 2.5 hours)**:
   - At 2:30 p.m., \( t = 2.5 \).
   - Calculate \( x \) and \( y \) at \( t = 2.5 \):
     - \( x = 520 \times 2.5 = 1300 \) miles.
     - \( y = 550 \times (2.5 - 1) = 550 \times 1.5 = 825 \) miles.
   - Calculate \( \frac{dx}{dt} \) and \( \frac{dy}{dt} \):
     - \( \frac{dx}{dt} = 520 \) (Plane A’s speed).
     - \( \frac{dy}{dt} = 550 \) (Plane B’s speed).

6. **Substitute into the Derivative**:
   \[
   \frac{dd}{dt} = \frac{(1300)(520) + (825)(550)}{\sqrt{1300^2 + 825^2}}
   \]

7. **Calculate the Values**:
   - Compute \( 1300^2 + 825^2 = 1690000 + 680625 = 2370625 \).
   - So, \( \sqrt{2370625} \approx 1539.5 \).
   - Now compute the numerator:
     \[
     (1300)(520) + (825)(550) = 676000 + 453750 = 1129750
     \]
   - Finally, divide to find \( \frac{dd}{dt} \):
     \[
     \frac{dd}{dt} = \frac{1129750}{1539.5} \approx 734.2 \, \text{mi/hr}
     \]

### Answer:
The rate at which the distance between the planes is changing at 2:30 p.m. is approximately **734.2 mi/hr**.

Would you like further explanation on any of these steps, or do you have additional questions?

---

### Related Questions to Explore:

1. What would the rate of change be if the planes had different speeds?
2. How does changing the angle of travel affect the rate at which the distance changes?
3. What would happen to the rate of change if one plane changed its direction mid-flight?
4. How can we apply similar principles to three-dimensional distance problems?
5. Can you find the maximum or minimum distance between two moving objects?

### Tip:
When dealing with related rates problems, always identify and differentiate the relevant formulae carefully before substituting values to avoid common calculation errors.

An airliner passes over an airport at noon traveling 520 mi/hr due west. At 1:00 p.m., another airliner passes over the same airport at the same elevation traveling due north at 550 mi/hr. Assuming both airliners maintain their (equal) elevations, how fast is the distance between them changing at 2:30 p.m.?

Learn how to solve a related rates problem involving two airplanes traveling in perpendicular directions at different speeds. Find the rate at which the distance between them is changing using the Pythagorean theorem and differentiation. Suitable for students in Grade 11-12 or introductory calculus.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation