Solve this problem: $80\%$ of the $40000$ fans in attendance will take the elevated train home. Find the exact time at which the last fan will have boarded their train. Fans start heading towards the train at $4$pm at a rate of $100$ per minute for one hour. At $5$pm the rate rises to $175$ per minute. At $6$pm the rate is $135$ per minute. To get on a train, fans must first walk into the station and go through the turnstiles. With no additional attendants, turnstiles process $60$ people per minute. Each additional attendant increases this rate by $40$ per minute.

Once on the platform, trains pick up passengers at an average rate of $120$ per minute since they carry approximately $1200$ people each and run every $10$ minutes.

Assume that only baseball fans board at Addison from $4$PM onward and that no fan is deterred by the long queues. At $5$pm they add $2$ additional attendants.

Your answer must be in the $24-$hour format $($for example, $18\mathrm{.}00$ and not $6$pm$).$ Round your final answer to $2$ decimals. Do not transform decimals in minutes $($for example, $18\mathrm{.}75$ and not $18$:$45).$