Yen and Mei independently embark on a multi-day open water swimming expedition along the same route. Each day, Yen and Mei begin swimming at 8 a.m. and swim continuously for 6 hours, with their distances recorded daily. Assume Yen and Mei remain at sea throughout the expedition and disregard their activities during the remaining 18 hours each day.
Let \(n\) represents the number of hours since 1 October at 8 a.m, where \(n=1\) corresponds to the first hour on 1 October, \(n=2\) corresponds to the second hour on 1 October, and so on.
Yen starts her journey on 1 October at 8 a.m. She completes 4 km in the first hour and the distance she covers in the next hour is 5% less than the previous hour. The distance continues to reduce in the same way for each subsequent day. Show that the distance covered in the first hour on 2 October is 2.94 km correct to 3 significant figures.[1]
Find the total distance Yen will swim by the end of 5 days.[2]
Find the theoretical total distance that Yen can swim over a long period of time.[2]
Due to unforeseen reasons, Mei starts her journey on 2 October at 8 a.m. She completes 5 km in the first hour and the distance she covers is 200 m less than the previous hour. The distance continues to reduce in the same way for each subsequent day. Show that the distance covered by Mei in the \(n\)th hour from 1 October at 8 a.m. is \(6.4-0.2n\), where \(n\geq 7\).[2]
Find the maximum total number of hours Mei can swim and the total distance covered by her.[4]
Find the smallest integer value of \(n\) for which the total distance covered by Mei exceeds the total distance covered by Yen.[2]