Answer the following with all steps, calculations, graphs as needed.

Problem $1$

The General Store at State University is an auxiliary bookstore located near the dormito$-$

ries that sells academic supplies, toiletries, sweatshirts and T$-$shirts, magazines, packaged

food items, and canned soft drinks and fruit drinks. The manager of the store has noticed

that several pizza delivery services near campus make frequent deliveries. The manager is

therefore considering selling pizza at the store. She could buy premade frozen pizzas and

heat them in an oven. The cost of the oven and freezer would be $$27,000.$ The frozen pizzas

cost $$3\mathrm{.}75$ each to buy from a distributor and to prepare $($including labor and a box$).$ To be

competitive with the local delivery services, the manager believes she should sell the pizzas

for $$8\mathrm{.}95$ apiece. The manager needs to write up a proposal for the university$$s director of

auxiliary services.

a$.$ Determine how many pizzas would have to be sold to break even.

b$.$ If the General Store sells $20$ pizzas per day, how many days would it take to break even?

c$.$ The manager of the store anticipates that once the local pizza delivery services start

losing business, they will react by cutting prices. If after a month $(30$ days$)$ the manager hasto lower the price of a pizza to $$7\mathrm{.}95$ to keep demand at $20$ pizzas per day, as she expects,

what will the new break$-$even point be$,$ and how long will it take the store to break even?

Problem $2$

A jewelry store makes necklaces and bracelets from gold and platinum. The store has $18$

ounces of gold and $20$ ounces of platinum. Each necklace requires $3$ ounces of gold and $2$

ounces of platinum, whereas each bracelet requires $2$ ounces of gold and $4$ ounces of plat$-$

inum. The demand for bracelets is no more than four. A necklace earns $$300$ in profit and a

bracelet, $$400.$ The store wants to determine the number of necklaces and bracelets to make

to maximize profit.

a$.$ Formulate a linear programming model for this problem.

b$.$ Solve this model by using graphical analysis.

Problem $3$

Universal Claims Processors processes insurance claims for large national insurance compa$-$

nies. Most claim processing is done by a large pool of computer operators, some of whom are

permanent and some of whom are temporary. A permanent operator can process $16$ claims

per day, whereas a temporary operator can process $12$ per day, and on average the company

processes at least $450$ claims each day. The company has $40$ computer workstations. A

permanent operator generates about $0\mathrm{.}5$ claim with errors each day, whereas a temporary

operator averages about $1\mathrm{.}4$ defective claims per day. The company wants to limit claims

with errors to $25$ per day. A permanent operator is paid $$64$ per day, and a temporary

operator is paid $$42$ per day. The company wants to determine the number of permanent

and temporary operators to hire to minimize costs.

a$.$ Formulate a linear programming model for this problem.

b$.$ Solve this model by using graphical analysis.