Final answer:
The optimal solution for minimizing faculty salaries while meeting all constraints is to offer 30 undergraduate courses and 30 graduate courses, as it satisfies the requirement of at least 60 total courses while minimizing costs.
Step-by-step explanation:
The student's question pertains to finding the optimal number of undergraduate and graduate courses to be offered in the fall so that the total cost of faculty salaries is minimized. This is a classical optimization problem that can be addressed using linear programming.
To formulate the problem, we identify the following constraints based on the information:
At least 30 undergraduate courses are required.
At least 20 graduate courses are required.
A minimum of 60 courses must be offered in total.
The cost function to minimize is:
Total Cost = $2,500(undergraduate courses) + $3,000(graduate courses)
Using these constraints and cost function, the college must find the number of undergraduate and graduate courses that minimizes the total cost while satisfying all constraints. The optimal solution will be where the lowest Total Cost is found without violating any of the constraints. In this case, it would be cost-effective to offer exactly 30 undergraduate courses and 30 graduate courses, satisfying the minimum requirement of 60 total courses while keeping the costs to a minimum, given the constraints.