$[0\mathrm{.}5/11\mathrm{.}5$ Points$]$

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$$AMMIMS$163.\mathrm{.}019.$

Better Products, Inc., manufactures three products used in the final assembly of solar energy cells on two machines. In a typical week, $40$ hours are available on each machine. The profit contribution and production time in hours per unit are as follows:

$\backslash$table$[[$Category$,$Product $1,$Product $2,$Product $3],[$Profit$/$unit$,$$$30,$$$50,$$ $ $20],[$Machine $1$ time$/$unit$,0\mathrm{.}5,2\mathrm{.}0,0\mathrm{.}75],[$Machine $2$ time$/$unit$,1\mathrm{.}0,1\mathrm{.}0,0\mathrm{.}5]]$

Two operators are required for machine $1-$thus, $2$ hours of labor must be scheduled for each hour of machine $1$ time. Only one operator is required for machine $2.$ A maximum of $100$ labor$-$hours is available for assignment to the machines during the coming week. Other production requirements are that product $1$ cannot account for more than $50\%$ of the units produced and that product $3$ must account for at least $20\%$ of the units produced.

$($a$)$ How many units of each product should be produced to maximize the total profit contribution? What is the projected weekly profit $($in $ $)$ associated with your solution?

Product $1$

Product $2$

Product $3$

weekly profit $ $$

$($b$)$ How many hours of production time will be scheduled on each machine?

machine $1$ hr

machine $2$ hr

$($c$)$ What is the value $($in $ $)$ of an additional hour of labor?

$