Final answer:
To find out how the distance between John and Chase is changing after Chase has been traveling for 3 hours, we can use the Pythagorean theorem. John will have traveled 210 miles west, and Chase will have traveled 165 miles north. The distance between them will be approximately 267 miles, and it will increase linearly as long as both continue at their respective constant speeds.
Step-by-step explanation:
To solve the problem of determining how the distance between John and Chase is changing after 3 hours after Chase begins traveling, we use the Pythagorean theorem. Since John and Chase are moving at right angles to each other, their paths form the two perpendicular sides of a right triangle, where the distance between them is the hypotenuse.
John starts heading west at 60 mph, and after 30 minutes, Chase starts heading north at 55 mph. When Chase starts, John has already traveled 30 miles (since 60 mph for half an hour yields 30 miles). After 3 hours, John will have traveled an additional 180 miles west (60 mph times 3 hours), making his total distance from the start point 210 miles. During the same time, Chase will have traveled 165 miles north (55 mph times 3 hours).
Using the Pythagorean theorem (a2 + b2 = c2), the distance between them (c) can be calculated:
c2 = 2102 + 1652
c = sqrt(2102 + 1652)
c ≈ sqrt(44100 + 27225)
c ≈ sqrt(71325)
c ≈ 267 miles
The rate of change of this distance with respect to time is not changing after 3 hours, as both are traveling at a constant speed, so the distance between them is increasing linearly as they continue to travel.