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In the next month, Technology Firm X will hire exactly \(24\) new engineers to at least partially fulfill the hiring needs of the firm's \(3\) engineering departments: the mechanical design department, which needs \(15\) new engineers; the electrical design department, which needs \(12\) new engineers; and the software design department, which needs \(8\) new engineers. Each of the engineering departments will receive at least \(1\) of the \(24\) new engineers, while none of the departments will receive more than it needs. Each of the new engineers will be employed in exactly one of the three departments. Assume that in every case in which a department \(\mathrm{D_1}\) needs more engineers than a department \(\mathrm{D_2}\), \(\mathrm{D_1}\) will receive more engineers than \(\mathrm{D_2}\).
Select the minimum number of engineers that the mechanical design department could receive, given this assumption and the information provided. And select the minimum number of engineers that the electrical design department could receive, given this assumption and the information provided. Make only two selections, one in each column.
Minimum received by mechanical design
Minimum received by electrical design
1
3
5
7
9
Let's create a table to organize our information:
| Department | Engineers Needed | Engineers Received | Constraints |
| Mechanical | 15 | M | \(\mathrm{M} \geq 1, \mathrm{M} \leq 15\) |
| Electrical | 12 | E | \(\mathrm{E} \geq 1, \mathrm{E} \leq 12\) |
| Software | 8 | S | \(\mathrm{S} \geq 1, \mathrm{S} \leq 8\) |
Key relationships:
We need to find:
These are separate optimization problems with the same constraints.
To minimize M, we need to maximize E and S while maintaining \(\mathrm{M} > \mathrm{E} > \mathrm{S}\).
Since \(\mathrm{S} \leq 8\) and \(\mathrm{E} > \mathrm{S}\), let's work backwards:
Can we make M smaller? If \(\mathrm{S} = 8\):
Minimum M = 9
To minimize E, we need to maximize M and minimize S while maintaining \(\mathrm{M} > \mathrm{E} > \mathrm{S}\).
Let's maximize \(\mathrm{M} = 15\) and see how small E can be:
If \(\mathrm{S} = 4\):
Can S be larger? If \(\mathrm{S} = 5\):
Minimum E = 5
Final Answer:
These values satisfy all constraints: