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A certain assistant professor in the Art History department at University X is being evaluated for promotion. One of the requirements is a performance score of at least \(\mathrm{4.0}\). The performance score is the weighted average of 3 component scores: one for research, one for teaching, and one for service, with the scores for research and teaching each weighted \(\mathrm{40\%}\) and the score for service weighted \(\mathrm{20\%}\). Each component score is between \(\mathrm{0.0}\) and \(\mathrm{5.0}\), inclusive.
Consistent with the given information, select for Minimum service score the least possible score the professor can receive for service and still achieve a performance score of at least \(\mathrm{4.0}\), and select for Minimum research score the least possible score the professor can receive for research and still achieve a performance score of at least \(\mathrm{4.0}\). Make only two selections, one in each column.
0
0.5
1
1.5
2
2.5
Since we're dealing with weighted components that sum to a total, let's use a simple equation format with a supporting table:
Performance Score Formula:
\(\mathrm{Performance\ Score} = 0.4(\mathrm{Research}) + 0.4(\mathrm{Teaching}) + 0.2(\mathrm{Service})\)
Component Breakdown:
| Component | Weight | Score Range |
| Research | 40% | 0.0 - 5.0 |
| Teaching | 40% | 0.0 - 5.0 |
| Service | 20% | 0.0 - 5.0 |
Target: \(\mathrm{Performance\ Score} \geq 4.0\)
We need to find TWO values:
To minimize one component, we need to maximize the others! This is because all components contribute positively to the total.
Strategy: Maximize research and teaching to allow service to be as low as possible.
Setting research = 5.0 and teaching = 5.0:
\(4.0 \leq 0.4(5.0) + 0.4(5.0) + 0.2(\mathrm{Service})\)
\(4.0 \leq 2.0 + 2.0 + 0.2(\mathrm{Service})\)
\(4.0 \leq 4.0 + 0.2(\mathrm{Service})\)
\(0 \leq 0.2(\mathrm{Service})\)
\(0 \leq \mathrm{Service}\)
Minimum service score = 0
Strategy: Maximize teaching and service to allow research to be as low as possible.
Setting teaching = 5.0 and service = 5.0:
\(4.0 \leq 0.4(\mathrm{Research}) + 0.4(5.0) + 0.2(5.0)\)
\(4.0 \leq 0.4(\mathrm{Research}) + 2.0 + 1.0\)
\(4.0 \leq 0.4(\mathrm{Research}) + 3.0\)
\(1.0 \leq 0.4(\mathrm{Research})\)
\(2.5 \leq \mathrm{Research}\)
Minimum research score = 2.5
Let's verify our minimum research score: