The production manager for the Whoppy soft drink company is considering the production of $2$ kinds of soft drinks: regular and diet. The company operates one $"8$ hour" shift per day. Therefore, the production time is $480$ minutes per day. During the production process, one of the main ingredients, syrup is limited to maximum production capacity of $675$ gallons per day. Production of a regular case requires $2$ minutes and $5$ gallons of syrup, while production of a diet case needs $4$ minutes and $3$ gallons of syrup. Profits for regular soft drink are $$3\mathrm{.}00$ per case and profits for diet soft drink are $$2\mathrm{.}00$ per case. The linear programming model and graph for this problem are provided below.

X$1=$ Number of Regular Cases to Produce

X$2=$ Number of Diet Cases to Produce

$Z=$ Profit

Maximize $Z=3x1+2x2$

Subject to

$2x1+4x2480$ mins of production

$51+32675$ gallons of syrup

$x1,x20$