Madison Musical Education Company $($MME$)$ provides instrumental music education to children of all ages. Payment for services comes from two sources: $(1)$ a contract with Country Day School to provide private music lessons for up to $150$ band students a year $($where a year is $9$ months of education$)$ for a fixed fee of $$160,000,$ and $(2)$ payment from individuals at a rate of $$100$ per month for $9$ months of education each year. In the $2008-2009$ school year, MME made a profit of $$5,000$ on revenues of $$295,000$ :

$1$

MME

Revenues:

Country Day School contract $$160,000$

Private students

$140,{000}_{?}$

Total revenues

$$300,000$

Expenses:

Administrative staff

Teaching staff

Facilities

Supplies

Total expenses

Profit

$291,{000}_{?}$

$$9,000$

MME conducted an activity analysis and found that teaching staff wages and supplies costs are variable with respect to student$-$months. $($A student$-$month is one student educated for one month.$)$ Administrative staff and facilities costs are fixed within the rage of $2,000-3,000$ studentmonths. At volumes between $3,000$ and $3,500$ student$-$months, an additional facilities charge of $$8,500$ would be incurred. During the last year, a total of $2,700$ student$-$months of education were provided, $1,395$ of which were for private students and $1,305$ of which were offered under the contract with Country Day School.

$(1)$ Compute the following using cost information from year $2008-2009$ operations: Fixed cost per year

Variable cost per student$-$month

$(2)$ Suppose that in $2009-2010$ Country Day School decreased its use of MME to $110$ students $($that is$,990$ student$-$months$).$ The fixed contract price of $$150,000$ was still paid. If everything else stayed as it was in $2008-2009,$ what profit or loss would be made in $20092010?$

$(3)$ Suppose that at the beginning of $2009-2010$ Country Day School decided not to renew its contract with MME, and the management of MME decided to try to maintain business as usual with only private students. How many students $($each signing up for $9$ months$)$ would MME require to continue to make a profit of $$9,000$ per year?