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Pottery Problem Part $1$

A company produces clay bowls and mugs. Two primary resources used in production are pottery clay and skilled labor.

Each bowl uses $1$ hour of labor and $4lb.$ of clay, and sells for $$50.$

Each mug uses $2$ hours of labor and $3lb.$ of clay, and sells for $$85.$

There are $40$ hours of labor and $120lb,$ of clay available each day. Labor cost is $$10$ per hour and clay cost is $$5$ per $lb.$ The company wants to know how many bowls and mugs to produce each day in order to maximize the total profit.

Formulate an LP model for this problem. $($Make sure your objective is a maximization of the total profit, not revenue. Also, make sure your LP model includes definitions of decision variables, an objective, constraints, and bounds on variables.$)$

Solve the problem using the graphical method.

Write the optimal solution and the optimal objective function value.

Write an interpretation of the optimal solution and the optimal objective function value.

Pottery Problem Part $2$

For the Pottery Problem, answer the following questions:

What would happen if bowls were selling for $$70$ per unit instead of $$50$ per unit?

What would happen if labor were a sunk cost, i$.$e$.,$ if we paid for $40$ hours per week whether we use them or not?

What would happen if there were an additional constraint that the number of bowls must be at least $50\%$ of the number of mugs?

Pottery Problem Part $3$