Final answer:
There is no combination of speeds at which the two runners will meet after 1 hour and 6 minutes.
Step-by-step explanation:
Let's assume that the speed of one athlete is x mph. Since the other athlete is running 2 mph faster, the speed of the second athlete would be (x + 2) mph.
If the distance between the runners is 11 miles, and they meet 1 hour and 6 minutes after starting, it means that the slower runner has been running for 1 hour and 6 minutes, while the faster runner has been running for 1 hour and 6 minutes minus the time it takes for the faster runner to return to the starting point.
Converting 1 hour and 6 minutes to hours, 1 hour and 6 minutes is equal to 1.1 hours.
Since the slower runner has been running for 1.1 hours at a speed of x mph, the distance the slower runner has covered is given by 1.1x miles.
On the other hand, the faster runner has been running for 1.1 hours minus the time it takes for the faster runner to return to the starting point at a speed of (x + 2) mph. Therefore, the distance the faster runner has covered is given by 1.1(x + 2) - 11 miles.
Since the two runners meet at some point, it means that the distance the slower runner has covered is equal to the distance the faster runner has covered. Therefore, we can equate the two distances:
1.1x = 1.1(x + 2) - 11
Simplifying the equation, we have:
1.1x = 1.1x + 2.2 - 11
Combining like terms, we get:
1.1x - 1.1x = -8.8
0 = -8.8
Since we have arrived at an invalid equation, it means that there is no solution to the problem. Therefore, there is no combination of speeds at which the two runners will meet after 1 hour and 6 minutes.