You and a friend are riding your bikes to a restaurant that you think is east; your friend thinks the restaurant is north. You both leave from the same point, with you riding at 16 mph east and your friend riding 12 mph north. After you traveled 4 mi, at what rate is the distance between you changing?

This problem involves related rates, where we need to determine the rate at which the distance between two moving objects is changing over time. We can use the Pythagorean theorem and derivatives to solve this.

### Step 1: Set up the coordinates
Let’s place your starting point at the origin \((0, 0)\) on a coordinate plane:
- You are riding east at a speed of 16 mph.
- Your friend is riding north at a speed of 12 mph.

After some time \(t\) hours:
- The distance you have traveled east is \(x(t) = 16t\) miles.
- The distance your friend has traveled north is \(y(t) = 12t\) miles.

### Step 2: Apply the Pythagorean theorem
The distance between you and your friend at time \(t\) is the hypotenuse of a right triangle, where the legs are the distances you and your friend have traveled east and north, respectively. Let \(d(t)\) represent the distance between you and your friend:

\[
d(t) = \sqrt{x(t)^2 + y(t)^2}
\]
\[
d(t) = \sqrt{(16t)^2 + (12t)^2} = \sqrt{256t^2 + 144t^2} = \sqrt{400t^2} = 20t
\]

### Step 3: Differentiate with respect to time
Now, to find the rate at which the distance between you and your friend is changing, we need to differentiate \(d(t)\) with respect to time \(t\):

\[
\frac{dd(t)}{dt} = \frac{d}{dt}(20t) = 20
\]

Thus, the distance between you and your friend is changing at a constant rate of **20 miles per hour**.

### Step 4: Find when \(t = \frac{4}{16}\) (since you have traveled 4 miles)
Since you traveled 4 miles east, the time \(t\) at that moment is:

\[
t = \frac{4}{16} = 0.25 \, \text{hours}.
\]

At this time, the rate at which the distance between you and your friend is changing remains 20 mph, as it is independent of \(t\).

### Final Answer:
The distance between you and your friend is changing at a rate of **20 mph** when you have traveled 4 miles.

Would you like more details or have any questions about the solution?

Here are 5 related questions to extend your understanding:
1. How does the rate of change in distance vary if your speed changes?
2. What would happen if both of you traveled in the same direction?
3. Can you solve for the distance between you and your friend after 0.5 hours?
4. How would the rate of change be affected if you traveled at 20 mph instead of 16 mph?
5. What if your friend traveled in a direction 30° east of north?

**Tip:** Related rates problems often rely on the Pythagorean theorem when distances are involved, especially with right triangles.

Learn how to solve a related rates problem involving two friends biking in perpendicular directions. The problem uses the Pythagorean theorem and differentiation to find the rate of change of the distance between them when one has traveled 4 miles. Suitable for Grades 10-12 students.

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