Problem $4.[15$ points$]$ Consider a company that produces two products, $\backslash ($ P$\_\{1\}\backslash )$ and $\backslash ($ P$\_\{2\}\backslash ).$ The profit per unit of $\backslash ($ P$\_\{1\}\backslash )$ is $\backslash (\backslash$$ $40\backslash )$ and the profit per unit of $\backslash ($ P$\_\{2\}\backslash )$ is $\backslash (\backslash$$ $30\backslash ).$ The production of these products requires two types of resources: Resource A and Resource B$.$ The availability and requirements are as follows:

Table $1$: Resource Requirements and Availability

The total availability of Resource A is $50$ units, and $80$ units for Resource B$.$

$($a$)$ Formulate a linear programming model for this problem and solve it graphically.

$($b$)$ Determine the shadow prices for Resource A and Resource B $.$

$($c$)$ What is the sensitivity range for resource A constraint? Please calculate the optimal objective value if the availability of resource A is changed to $100$ units.