C$-$town brewery brews two beers: Expansion Draft and Burning River.

Expansion Draft sells for $$20$ per barrel, while Burning River sells for $$8$ per barrel.

Producing a barrel of Expansion draft takes $8$ pounds of corn and $4$ pounds of hops. Producing a barrel of Burning River requires $2$ pounds of corn, $6$ pounds of rice, and $3$ pounds of hops. The brewery has $500$ pounds of corn, $300$ pounds of rice, and $400$ pounds of hops.

Lady Giga is a marketing manager at C$-$town brewery and she started to formulate the above problem using Excel to maximize C$-$town$$s revenue. Her worksheet and solver entries can be found below.

$($a$)$ What is the Excel formula in Cell C$12?($What did Lady Giga type in Cell C$12?)$

$($b$)$ What is the Excel formula in Cell C$15?($What did Lady Giga type in Cell C$15?)$

After clicking on $$Solve$,$ the optimal decision variables turn out to be $50$ and $50,$ respectively. Lady Giga generated the following sensitivity report:

Variable Cells

Final Reduced Objective Allowable Allowable Cell Name Value Cost Coefficient Increase Decrease

$C$$2$ Expansion Draft $50020$

$12$

$20$ $D$$2$ Burning River

$500$

$81$E$+30$

$3$

Constraints Final Shadow Constraint Allowable Allowable

Cell Name Value Price R$.$H$.$ Side Increase Decrease

$C$$15$ Corn Used $500$

$2\mathrm{.}5500100400$ $C$$16$ Rice Used $300$

$0\mathrm{.}5300150$

$300$ $C$$17$ Hops Used

$350$

$0$

$4001$E$+30$

$50$

$($c$)$ What is the optimal objective value?

$($d$)$ If C$-$town brewery has $120$ additional pounds of corn, by how much will the optimal revenue

increase?

$($e$)$ If C$-$town brewery lost $120$ pounds of rice, by how much will the optimal revenue change?

$($f$)$ If C$-$town brewery has $120$ additional pounds of hops, by how much will the optimal revenue

increase?

$($g$)$ If the unit revenue for Expansion Draft is increased by $$5,$ how much will the optimal revenue

increase?