A family$-$run business produces hand$-$crafted bowls and mugs from clay for sale at an artisans

market. The two resources needed to produce these goods are pottery clay and skilled labor. Given

these limited resources, the business wishes to know how many bowls and mugs to produce each

week. Each bowl requires $400$ grams of clay and $1$ hour of labor, and yields a profit of $$10.$ Each mug

requires $300$ grams of clay and $1\mathrm{.}2$ hours of labor, and yields a profit of $$8.$ There are $30$ labor hours

and $8\mathrm{.}4$ kilograms of clay available each week for production. Formulate the linear programming

model to determine the product mix that will maximize the business$$s weekly profit. Make sure

to define the decision variables, the objective function, and all necessary constraints. Provide all

constraints in standard form. You do not need to solve the LP$.$