Funkytown brewery brews two types of beers: Draught and River. Draught sells for $$20$ per barrel and River sells for $$12$ per barrel. The products are sold as full barrels. Producing a barrel of Draught takes $8$ pounds of corn and $4$ pounds of hops. Producing a barrel of River requires $2$ pounds of corn, $5$ pounds of rice, and $4$ pounds of hops. The brewery has $500$ pounds corn, $300$ pounds of rice, and $400$ pounds of hops. Funkytown wants to know the optimal mix of products that maximizes revenues. You are hired by Funkytown to develop a linear programming model$\mathrm{.}4$a$)(10$ points$)$ What are the decision variables? How many decision variables are there? b$)(5$ points$)$ What is the objective function? Explain clearly and write the mathematicalexpression$($s$).$ c$)(15$ points$)$ What are the constraints? Explain clearly and write the mathematicalexpression$($s$).$ d$)(3$ points$)$ What are the sign and type restrictions? Explain clearly and write themathematical expression$($s$).$