Recall the production model from Section $1\mathrm{.}3$:

Max $10x$

$s.t.5x40$

$,x0$

Suppose the firm in this example considers a second product that has a unit profit of $$ and requires $2$ hours for each unit produced. Assume total production capacity remains $40$ units. Use $y$ as the number of units of product $2$ produced.

a$.$ Show the mathematical model when both products are considered simultaneously,

b$.$ Identify the controllable and uncontrollable inputs for this model.

c$.$ Draw the flowchart of the input$-$output process for this model $($see Figure $1\mathrm{.}5).$

d$.$ What are the optimal solution values of $x$ and $y?$