Two athletes are to run \(20\text{ km}\) by running \(50\) laps around a circular track of length \(400\text{ m}\). They aim to complete the distance in between \(1\frac12\) hours and \(1\frac34\) hours inclusive.
Athlete \(A\) runs the first lap in \(T\) seconds and each subsequent lap takes \(2\) seconds longer than the previous lap. Find the set of values of \(T\) which will enable \(A\) to complete the distance within the required time interval.[4]
Athlete \(B\) runs the first lap in \(t\) seconds and the time for each subsequent lap is \(2\%\) more than the time for the previous lap. Find the set of values of \(t\) which will enable \(B\) to complete the distance within the required time interval.[4]
Assuming that each athlete completes the \(20\text{ km}\) run in exactly \(1\frac12\) hours, find the difference in the athletes' times for their \(50\)th laps, giving your answer to the nearest second.[3]