Problem Solving - A manufacturer of cell phones makes a profit of $25 on a deluxe model and $30 on a standard model. The company wishes to produce at least 80 deluxe models and at least 100 standard models per day. To maintain high quality, the daily production should not exceed 200 phones. How many of each type should be produced daily in order to maximize the profit? HOW DO I WRITE THE CONSTRAINTS TO GRAPH A FEASIBLE REGION?
deluxe model -- x
standard model -- y
x ≥ 80
y ≥ 100
x+y < 200
Profit = 25x + 30y
To graph the feasible region, you need to write the constraints based on the given information. In this case, the constraints are:
1. Profit constraint for deluxe models: The manufacturer wants to produce at least 80 deluxe models per day, which means the number of deluxe models should be greater than or equal to 80.
In equation form: x ≥ 80, where x represents the number of deluxe models.
2. Profit constraint for standard models: The manufacturer wants to produce at least 100 standard models per day, which means the number of standard models should be greater than or equal to 100.
In equation form: y ≥ 100, where y represents the number of standard models.
3. Production capacity constraint: The manufacturer wants to produce no more than 200 phones per day. The total number of phones produced, which is the sum of deluxe and standard models, should be less than or equal to 200.
In equation form: x + y ≤ 200.
4. Non-negativity constraint: The number of phones produced cannot be negative. Both the deluxe and standard models should be greater than or equal to 0.
In equation form: x ≥ 0 and y ≥ 0.
Once you have these constraints, you can plot them on a graph by converting them to inequalities. The feasible region will be the area on the graph where all the constraints are satisfied, representing solutions that meet the given conditions.