The maximum income that Dr. Lum can earn is $24,600 by teaching 3 classes at Hilltop College and 2 classes at Serra College.
To maximize his income, Dr. Lum needs to determine the number of classes he should teach at each college. Let x be the number of classes he teaches at Hilltop College and y be the number of classes he teaches at Serra College. Then, the objective function that Dr. Lum wants to maximize is:
Income = 4000x + 4200y
The constraints are:
x + y ≤ 5 (Dr. Lum can teach up to 5 classes per semester)
3x + 4y ≤ 18 (Dr. Lum cannot spend more than 18 hours per week preparing lessons and grading papers)
Since Dr. Lum cannot teach a negative number of classes, x and y must be non-negative. Therefore, the complete linear programming problem is:
Maximize: Income = 4000x + 4200y
Subject to: x + y ≤ 5
3x + 4y ≤ 18
x ≥ 0, y ≥ 0
To solve this problem graphically, we can plot the two constraint lines on a graph and shade the feasible region that satisfies both constraints. The feasible region is the region that is below both lines and in the first quadrant. The optimal solution is the point in the feasible region that maximizes the objective function.
Solving the system of equations, we get the coordinates of the vertices of the feasible region:
(0, 0), (0, 4.5), (3, 2), (5, 0)
Evaluating the objective function at each vertex, we get:
(0, 0): Income = 0
(0, 4.5): Income = 18900
(3, 2): Income = 24600
(5, 0): Income = 20000
Therefore, the maximum income that Dr. Lum can earn is $24,600 by teaching 3 classes at Hilltop College and 2 classes at Serra College.