question $3$ problem $2$

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, consider the following questions $($separately from the others$)$:

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 day 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 $3$

In addition to bowls and mugs, the company now wants to produce clay jugs.

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.$

Each jug uses $1\mathrm{.}5$ hours of labor and $5lb.$ of clay and sells for $$100.$

As before, 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, mugs, and jugs to produce each

day in order to maximize the total profit.

Formulate an LP model for this problem.

Can you solve this problem using the graphical method?