Football Fanatics, Inc. makes three types of footballs $$ the Pee Wee, the College Canon, and the Pro Player. The labor and materials requirements for each football are shown in the table below with information regarding the maximum amount of labor hours available $($for a given activity$)$ and the maximum amount of each material available.

Labor $($hours$$football$)$ Material Requirements $($gram$$football$)$

Revenue Cut Assemble Rubber Leather Stitching

Pro Player $550\mathrm{.}30\mathrm{.}525020060$

College Canon $480\mathrm{.}20\mathrm{.}422518050$

Pee Wee $300\mathrm{.}10\mathrm{.}315010040$

Max Available $4010035,00028,0005,000$

Cost issues $15200\mathrm{.}020\mathrm{.}050\mathrm{.}03$

$[$$$$hour$][$$$$gram$]$

The firm purchases only as much of the three materials $($rubber$,$ leather, and stitching$)$ as it uses at the prices shown in the table. It wants the Pro Player to comprise at least $10\%$ of all the footballs it produces and sells. In addition, the number of Pro Player and College Canon footballs $($in total$)$ must not be more than $50$ percent of the total number of footballs produced and sold.

$(1)$ How many of each type of football should be produced to maximize its profits, subject to the availability and sales$-$planning constraints given? $($fractional answers are OK$)$

$(2)$ How does this result differ when integer decisions $($i$.$e$.,$ whole numbers of footballs$)$ are assumed necessary for planning estimates? Provide the new decisions and the final profit.

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