Final answer:
Three hours later, the distance between car A and car B is increasing at a rate of approximately 39.2 mph. This is determined using the Pythagorean theorem and the concept of related rates.
Step-by-step explanation:
To determine the rate at which the distance between two cars is changing after 3 hours, we will use the Pythagorean theorem and the concept of related rates.
We're essentially dealing with a right triangle where the legs represent the distances traveled by each car from the point of intersection, and the hypotenuse represents the distance between the two cars.
After 3 hours, car A would have traveled 3 hours × 65 miles per hour = 195 miles, and it would now be at the intersection.
Car B would have traveled 3 hours × 40 miles per hour = 120 miles west of the intersection.
Let x be the distance car B is from the intersection, and y be the distance car A is from the intersection.
Let d be the distance between the two cars.
After 3 hours, x = 35 miles + 120 miles and y = 215 miles - 195 miles. So we have x = 155 miles and y = 20 miles.
Using the Pythagorean theorem: d² = x² + y².
To find the rate at which d changes, we differentiate both sides of the equation with respect to time (t), which gives us: 2d(dd/dt) = 2x(dx/dt) + 2y(dy/dt).
Since car B is moving away from the intersection (west), dx/dt is positive 40 mph, and since car A is now at the intersection, dy/dt is 0.
Solving for dd/dt we get dd/dt = (x × dx/dt) / d. Plugging in the values we get dd/dt = (155 miles × 40 mph) / (the square root of (155² + 20²) miles).
Performing the calculations, dd/dt comes out to approximately 39.2 miles per hour.
Thus, three hours later, the distance between the two cars is increasing at a rate of about 39.2 mph.