Jorgenson Company produces two types of Gears, X and Y$,$ with unit contribution margins of $$25$ and $$10,$ respectively.$$ Each gear must be notched by a special machine.$$ The company owns $8$ machine that together provide $40,000$ hours of machine time per year.$$ Gear X requires $2$ hours of machine time, and Gear Y requires $\mathrm{.}5$ hour of machine time.$$ There are no other constraints.

$1.$ What is the contribution margin per hour of machine time each year?

$2.$ Given the information, calculate the number of units of each product $($product mix$)$ that the company should make.$$ Provide support for these values.

$3.$ What is the resulting contribution margin of the optimal product mix?

$4.$ Now let's assume that there are two constraints for this company, the same constraint of $40,000$ hours of machine time available per year and now a demand constraint of a maximum $60,000$ units of each gear that can be sold.$$ What is the new optimal product mix?

$5.$ What is the resulting contribution margin of the optimal product mix?