Solve the following linear programming problems by the graphical method
Dr. Lum teaches part-time at two different community colleges, Hilltop College and Serra College. Dr. Lum can teach up to 5 classes per semester. For every class taught by him at Hilltop College, he needs to spend 3 hours per week preparing lessons and grading papers, and for each class at Serra College, he must do 4 hours of work per week. He has determined that he cannot spend more than 18 hours per week preparing lessons and grading papers. If he earns $4,000 per class at Hilltop College and $5,000 per class at Serra College, how many classes should he teach at each college to maximize his income, and what will be his income?
so, what are the constraints, and the value to maximize?
In algebra, not words.
He can only teach 5 classes and he can't spend more than 18 hours per week preparing lessons and grading papers-Those are the constraints I think and he has to maximize his income
To solve this linear programming problem by the graphical method, we need to set up the constraints and the objective function. Let's assign the following variables:
Let x = the number of classes taught at Hilltop College
Let y = the number of classes taught at Serra College
The objective function is the income, which is given by:
Income = 4000x + 5000y
Now, let's set up the constraints:
1. The total number of classes taught should be less than or equal to 5:
x + y ≤ 5
2. The total number of hours spent on preparing lessons and grading papers should be less than or equal to 18:
3x + 4y ≤ 18
3. The number of classes taught at each college should be non-negative:
x ≥ 0
y ≥ 0
Now, let's plot these constraints on a graph:
First, let's plot the constraint x + y ≤ 5:
- Plot the line x + y = 5.
- Shade the region below this line.
Next, let's plot the constraint 3x + 4y ≤ 18:
- Plot the line 3x + 4y = 18.
- Shade the region below this line.
Finally, let's plot the non-negativity constraints:
- Shade the region in the first quadrant of the graph.
The feasible region, which is the shaded region where all the constraints are satisfied, is the region where Dr. Lum can choose the number of classes taught at each college.
To find the optimal solution, we need to find the corner points of the feasible region and calculate the income for each corner point. The corner point with the highest income will be the solution.
Once you have plotted the graph and found the corner points, please provide the coordinates of the corner points to proceed further.
To solve this linear programming problem using the graphical method, we will follow these steps:
Step 1: Define the decision variables.
Let's denote the number of classes that Dr. Lum teaches at Hilltop College as x and the number of classes he teaches at Serra College as y.
Step 2: Formulate the objective function.
The objective is to maximize Dr. Lum's income. His income can be calculated using the following equation:
Income = 4000x + 5000y
Step 3: Formulate the constraints.
We have two constraints in this problem:
1. The total number of classes taught should not exceed 5 per semester:
x + y ≤ 5
2. The total hours spent preparing and grading papers should not exceed 18 hours per week:
3x + 4y ≤ 18
Step 4: Graph the constraints.
To graph the constraints, we first convert them into linear equations and then plot them on a graph.
For the first constraint:
x + y ≤ 5
Rearranging the equation, we get:
y ≤ 5 - x
Now, let's plot this line on the graph.
For the second constraint:
3x + 4y ≤ 18
Rearranging the equation, we get:
4y ≤ 18 - 3x
y ≤ (18 - 3x)/4
Now, let's plot this line on the graph as well.
Step 5: Determine the feasible region.
The feasible region is the area on the graph where both constraints are satisfied, represented by the shaded region.
The feasible region is the intersection of the shaded region by both constraints.
Step 6: Evaluate the objective function at the vertices.
The vertices of the feasible region are the points where the lines intersect. We evaluate the objective function (income equation) at these points to determine which one gives the maximum income.
Step 7: Determine the maximum income and the number of classes.
By evaluating the income equation at each vertex, we can find the point that gives the maximum income. The corresponding x and y values will tell us the number of classes to teach at each college.
After following these steps, you will be able to determine the number of classes he should teach at each college to maximize his income and the actual income.