Excel Corporation manufactures three products at its plant. The plant capacity is limited to $120,000$ machine hours per year on of single$-$shift basis. Direct material and direct labor costs are variable.

The following data are available for planning purposes:

$\backslash$table$[[,,,$Direct,,Variable,$],[,$Total,,,,,$],[,$Unit,,Materials,Direct,Overhead,Machine$],[,$Demand,Sales,,Labor,,Hours$],[,$For,Price,Cost Per,Cost,Cost Per,Per$],[,$Next,Per,,Per,,$],[,$Year,Unit,Unit,Unit,Unit,Unit$],[$XL$1,200,000,$$$10\mathrm{.}00,$$$4\mathrm{.}00,$$$2\mathrm{.}00,$$$2\mathrm{.}00,0\mathrm{.}20],[$XL$2,200,000,$$$14\mathrm{.}00,$$$4\mathrm{.}70,$$$3\mathrm{.}30,$$$3\mathrm{.}30,0\mathrm{.}35],[$XL$3,200,000,$$$12\mathrm{.}00,$$$5\mathrm{.}00,$$$2\mathrm{.}50,$$$2\mathrm{.}50,0\mathrm{.}25]]$

$($a$)$ Given the capacity constraint, determine the production levels for the three products that will maximize profits. Round down to nearest unit.

XL$1$

XL$2$

XL$3$

Round down to nearest unit.

$($b$)$ If the company authorizes overtime in order to produce more units of XL$2,$ the direct labor