Final answer:
To minimize costs, Memorial Hospital should hire 24 nurses and 12 aides based on the provided constraints and goal of cost minimization. This is not listed in the options provided; hence, there might be an error in the given choices. The optimal solution is determined using linear programming techniques.
Step-by-step explanation:
The question involves finding the optimal combination of nurses and nurse's aides to hire at Memorial Hospital to minimize costs while adhering to specific constraints. To solve this, we set up a linear programming problem with the following considerations:
- Minimum number of staff needed: 20
- Maximum number of staff that can be hired: 50
- Minimum number of aides needed: 12
- Number of nurses must be at least twice the number of aides
- Cost of hiring a nurse is $35,000 per year
- Cost of hiring an aide is $18,000 per year
Formulating this as a linear algebra problem, we let N represent the number of nurses and A represent the number of aides. We want to minimize the cost function C = 35000N + 18000A, subject to the constraints:
- N + A ≤ 50
- N + A ≥ 20
- A ≥ 12
- N ≥ 2A
To minimize costs, the hospital should hire the minimum number of aides required (12) and the minimum number of nurses required to be at least twice that amount (24), leading to the answer:
24 nurses and 12 aides
This isn't one of the provided options, so there may be an error in the options given. However, using the constraints and goal of minimizing costs, this is the optimal solution.