Question

COLLEGE ADMINISTRATION A small college has full, associate, and assistant professors. Each week, full professors teach 6 hours, spend 8 hours grading papers, and spend 4 hours doing committee work. Associate professors teach 9 hours each week, spend 12 hours grading papers, and 3 hours doing committee works, while assistant professors teach 12 hours a week, spend 15 hours grading, and 1 hour doing committee work a. Develop a method involving matrix inversion for finding the number of professors of each rank that are needed if the college requires a total of 7 hours of teaching each week, G hours of grading, and C hours of committee work. b. The administration is considering two plans that include different requirements for faculty activity in weekly hours) as listed in the table below: Teaching Grading Committee work Plan #1 1,227 1,572 265 Plan #2 1.305 1.680 315 We Use the procedure you developed in part (a) to find the number of professors of each rank that should be retained by the college to facilitate each plan.

Answer 1

To find the number of professors of each rank that are needed, a method involving matrix inversion can be used. The system of equations representing the teaching, grading, and committee work hours can be written in matrix form and solved using the inverse of the coefficient matrix.

Step-by-step explanation:

To find the number of professors of each rank needed, we can set up a system of equations using matrix inversion. Let's define the variables:

• F = number of full professors
• A = number of associate professors
• As = number of assistant professors

The given information can be represented by the following equations:

• Teaching hours: 6F + 9A + 12As = 7
• Grading hours: 8F + 12A + 15As = G
• Committee work hours: 4F + 3A + As = C

Using matrix inversion, we can write the equations in matrix form:

[6 9 12][F A As] = [7 G C]

To solve for F, A, and As, we can multiply both sides by the inverse of the coefficient matrix:

[F A As] = [6 9 12]^-1 [7 G C]

Once we have the inverse of the coefficient matrix, we can substitute in the values for 7, G, and C to find the values of F, A, and As.