$[12$ marks$]$ The population y$($t$)$ of an ant colony in an ant farm can be modelled by the differential equation $($dy$)/($dt$)=$Ky where t is in months and the growth constant K is $0\mathrm{.}1.$ When the ant population reaches $1,000,$ a researcher begins to remove ants from the ant colony at a rate of $20$ ants in the first month, $40$ ants in the second, $60$ in the third and so on$.($a$)$ Explain why a new differential equation that models the ant population taking into account the regular removal of ants is $($dy$)/($dt$)=0\mathrm{.}1$y$-20$t What is the appropriate initial condition for this differential equation? $($d$)$ Is this rate of 'harvesting' ants sustainable, or will the colony size eventually become zero? $($b$)$ Show that the ant population at time t is y$($t$)=200$t$+2000-1000$e$^(0\mathrm{.}1$t$).($c$)$ After $5$ months, how many ants will there be in the colony? I need a$,$ b$,$ c$,$ d all answers. I give upvote... Thanks advance. No ai$...$ I need hand written notes. If it's possible..