$19.$ Modeling Cabinet Produc$1$on$.$ Brooklyn Cabinets is a manufacturer of kitchen cabinets. The two cabinetry

styles manufactured by Brooklyn are contemporary and farmhouse. Contemporary style cabinets sell for $$90$

and farmhouse style cabinets sell for $$85.$ Each cabinet produced must go through carpentry, pain$=$ng$,$ and

finishing processes. The following table summarizes how much $=$me in each process must be devoted to each

style of cabinet.

Style Carpentry

Hours per Process

Painting Finishing

Contemporary $2\mathrm{.}01\mathrm{.}51\mathrm{.}3$

Farmhouse $2\mathrm{.}51\mathrm{.}01\mathrm{.}2$

Carpentry costs $$15$ per hour, pain$=$ng costs $$12$ per hour, and finishing costs $$18$ per hour; and the weekly

number of hours available in the processes is $3000$ in carpentry, $1500$ in pain$=$ng$,$ and $1500$ in finishing.

Brooklyn also has a contract that requires the company to supply one of its customers with $500$ contemporary

cabinets and $650$ farmhouse style cabinets each week.

Define x and y as follows. LO $4$

x $=$ the number of contemporary style cabinets produced each week

y $=$ the number of farmhouse style cabinets produced each week

a$.$ Develop the objec$=$ve func$=$on$,$ assuming that Brooklyn Cabinets wants to maximize the total weekly

profit.

b$.$ Show the mathema$=$cal expression for each of the constraints on the three processes.

c$.$ Show the mathema$=$cal expression for each of Brooklyn Cabinets$$ contractual agreements.