Alice has a tire manufacturing company that can produce a maximum of $4000$ tires per annum. The fixed costs per annum are $$70,000.$ The variable costs are $$50$ per tire, and the selling price per unit is $$100.$

$($i$)$ Calculate the number of tires she should produce and sell per year to break$-$even.

$($ii$)$ In $2008,$ Alice produced and sold $1000$ tires. Did she make a profit or incur a loss, and by what amount?

$($iii$)$ In $2009,$ Alice produced and sold $3000$ tires. Did she make a profit or incur a loss, and by what amount?

$($iv$)$ In $2010,$ the manufacturing plant operated at $90\%$ of capacity to produce and sell tires. Calculate the profit made or loss incurred.

$($v$)$ In $2011,$ Alice wanted a profit of $$60,000.$ Find the number of tires that she would have to produce$/$sell$,$ and the level?

$($vi$)$ What is the maximum profit per annum that she can expect from this facility?