$18.$ The following is a linear programming model for analyzing the product mix of Maxine's Hat Company, which produces three hat styles:

Maximize: $ $7$ x$\_1+$$ $8$ x$\_2+$$ $6$ x$\_3=$Z

Subject to: $2$ x$\_1+4$ x$\_2+2$ x$\_3<=120($machine A time$)$

$5$ x$\_1+3$ x$\_2+2$ x$\_3<=400($machine B time$)$

$2$ x$\_1+2$ x$\_2+4$ x$\_3<=110($machine C time$)$

x$\_1>=0,$ x$\_2>=0,$ and x$\_3>=0$

The POM for Windows printout in Figure D$\mathrm{.}15!$ shows the optimal solution to the problem. Figure D$\mathrm{.}15$ Solution Screens for Maxine's Hat Company Consider each of the following statements independently, and state whether it is true or false. Explain each answer. a$.$ If the price of hat $3$ were increased to $ $11\mathrm{.}50,$ it would be part of the optimal product mix. Hint: Hat $3$ is represented by x$\_3$ and its optimal value currently is $0,$ which means that it is not to be produced $($and not part of the optimal product mix$).$ b$.$ The capacity of machine B can be reduced to $280$ hours without affecting profits. c$.$ If machine C had a capacity of $115$ hours, the production output would remain unchanged.