An electronics company is considering which research and development projects to select. The following are the resource requirements and the estimated profit for each project:

Project Expense $($$$)$ Scientists Required Estimated Profit $($$$)$

$170,0007360,000$

$2110,0009820,000$

$350,0008290,000$

$445,0004160,000$

$590,0007560,000$

$685,0006610,000$

$770,0008480,000$

$865,0005410,000$

The number of available scientists is $40$ and the budget available for these research and development projects is $$300,000.$

Furthermore, if project $2$ is selected, project $5$ must also be selected $($but not vice versa$).$ Projects $3$ and $4$ are mutually exclusive $($both cannot be selected$).$

Formulate an integer programming model to determine which projects should be selected to maximize profit.