$4.$ Consider the linear program in Problem $3.$ The value of the optimal solution is $48.$ Suppose that the right$-$hand side for constraint $1$ is increased from $9$ to $10.$

a$.$ Use the graphical solution procedure to find the new optimal solution.

b$.$ Use the solution to part $($a$)$ to determine the dual value for constraint $1.$

c$.$ The computer solution for the linear program in Problem $3$ provides the following right$-$ hand$-$side range information:

What does the right$-$hand$-$side range information for constraint $1$ tell you about the dual

value for constraint $1?$

d$.$ The dual value for constraint $2$ is $3.$ Using this dual value and the right$-$hand$-$side range

information in part $($c$),$ what conclusion can be drawn about the effect of changes to the right$-$hand side of constraint $2?$