Solve the following linear programming model graphically:

Solve the following linear programming model graphically:

Gillian$$s Restaurant has an ice$-$cream counter where it sells two main products, ice cream and frozen yogurt, each in a variety of flavors. The restaurant makes one order for ice cream and yogurt each week, and the store has enough freezer space for $115$ gallons total of both products. A gallon of frozen yogurt costs $$0\mathrm{.}75,$ and a gallon of ice cream costs $$0\mathrm{.}93,$ and the restaurant budgets $$90$ each week for these products. The manager estimates that each week the restaurant sells at least twice as much ice cream as frozen yogurt. Profit per gallon of ice cream is $$4\mathrm{.}15,$ and profit per gallon of yogurt is $$3\mathrm{.}60.$

Formulate a linear programming model for this problem.

Solve this model by using graphical analysis.

In Problem $30,$ how much additional profit would the restaurant realize each week if it increased its freezer capacity to accommodate $20$ extra gallons total of ice cream and yogurt?

Copperfield Mining Company owns two mines, each of which produces three grades of ore$$high$,$ medium, and low. The company has a contract to supply a smelting company with at least $12$ tons of high$-$grade ore, $8$ tons of medium$-$grade ore, and $24$ tons of low$-$grade ore. Each mine produces a certain amount of each type of ore during each hour that it operates. Mine $1$ produces $6$ tons of high$-$grade ore, $2$ tons of medium$-$grade ore, and $4$ tons of low$-$grade ore per hour. Mine $2$ produces $2,2,$ and $12$ tons, respectively, of high$-,$ medium$-,$ and low$-$grade ore per hour. It costs Copperfield $$200$ per hour to mine each ton of ore from mine $1,$ and it costs $$160$ per hour to mine each ton of ore from mine $2.$ The company wants to determine the number of hours it needs to operate each mine so that its contractual obligations can be met at the lowest cost.

Formulate a linear programming model for this problem.

Solve this model by using graphical analysis.

A canning company produces two sizes of cans$$regular and large. The cans are produced in $10,000-$can lots. The cans are processed through a stamping operation and a coating operation. The company has $30$ days available for both stamping and coating. A lot of regular$-$size cans requires $2$ days to stamp and $4$ days to coat, whereas a lot of large cans requires $4$ days to stamp and $2$ days to coat. A lot of regular$-$size cans earns $$800$ profit, and a lot of large$-$size cans earns $$900$ profit. In order to fulfill its obligations under a shipping contract, the company must produce at least nine lots. The company wants to determine the number of lots to produce of each size can in order to maximize profit.

Formulate a linear programming model for this problem.

Solve this model by using graphical analysis.

A manufacturing firm produces two products. Each product must undergo an assembly process and a finishing process. It is then transferred to the warehouse, which has space for only a limited number of items. The firm has $80$ hours available for assembly and $112$ hours for finishing, and it can store a maximum of $10$ units in the warehouse. Each unit of product $1$ has a profit of $$30$ and requires $4$ hours to assemble and $14$ hours to finish. Each unit of product $2$ has a profit of $$70$ and requires $10$ hours to assemble and $8$ hours to finish. The firm wants to determine the quantity of each product to produce to maximize profit.

Formulate a linear programming model for this problem.

Solve this model by using graphical analysis.

Assume that the objective function in Problem $34$ has been changed from to Determine the slope of each objective function, and discuss what effect these slopes have on the optimal solution.

The Valley Wine Company produces two kinds of wine$$Valley Nectar and Valley Red. The wines are produced from $64$ tons of grapes the company has acquired this season. A $1,000-$gallon batch of Nectar requires $4$ tons of grapes, and a batch of Red requires $8$ tons. However, production is limited by the availability of only $50$ cubic yards of storage space for aging and $120$ hours of processing time. A batch of each type of wine requires $5$ cubic yards of storage space. The processing time fo