Final answer:
Ariel's brother catches up to Ariel at 5 p.m. after starting two hours later but driving 15 mph faster, thus closing the initial 90-mile gap between them.
Step-by-step explanation:
The problem at hand is a relative motion problem in which we calculate the time it takes for Ariel's brother to catch up to her, given the start times and speeds of both individuals. First, we determine how far Ariel has driven by the time her brother starts. Ariel leaves at 9 a.m. and drives for 2 hours at a speed of 45 mph before her brother starts chasing after her at 11 a.m. The distance Ariel has covered by 11 a.m. is 45 mph multiplied by 2 hours, which equals 90 miles.
Next, we determine how quickly her brother is closing the gap. Ariel's brother is traveling at 60 mph, which is 15 mph faster than Ariel's 45 mph. The time it takes for Ariel's brother to catch up to her is the distance Ariel has traveled divided by the difference in their speeds. So, the catch-up time is 90 miles divided by 15 mph, which equals 6 hours.
Since Ariel's brother started at 11 a.m., to find the catch-up time, we add the 6 hours to 11 a.m., which results in 5 p.m. Therefore, Ariel's brother catches up to Ariel at 5 p.m.