The Vancouver Fraser Port Authority is planning out their internships for the next calendar year. They would like to have $3$ student interns working from Jan$-$Apr, $5$ students working from May$-$Aug, and $4$ students working from Sep$-$Dec. They can hire students for $4-$month internships $($Jan$-$Apr, May$-$Aug, Sep$-$Dec$)$ and they can also hire students for $8-$month internships $($Jan$-$Aug, May$-$Dec$).$ In total, they would like to hire at least $8$ student interns.The salary for a $4-$month internship is $$14,000$ and for an $8-$month internship is $$28,000.$ There are also additional hiring costs associated with each type of internship which differ based on time of year $($recruitment$,$ interviewing, checking references, setting up payroll, etc.$)$

At the optimal solution, what is the total cost of these internships $($including salaries and hiring costs$).$

Recruiting for Jan$-$Apr internships is becoming really really difficult!$$Suppose the full cost $($salary $+$ hiring costs$)$ for a $4-$month Jan$-$Apr internship increases by a whopping $$176,000.$With this change, how many students should the Port Authority hire for $4-$month Jan$-$Apr internships?

OK$,$ forget about the above change to the cost of Jan$-$Apr internships. Suppose instead that the full cost of a May$-$Aug internship decreases by $$1,271.$What is the new total cost of this model?

Again forget the above change. Now suppose instead that $2$ more students are needed from May$-$Aug.What is the new total cost of this model?