A television manufacturer makes console and wide-screen televisions. The profit per unit is $125 for the console televisions and $200 for the wide-screen televisions.
Write the objective function that describes the total monthly profit.
Write a system of three inequalities that describes the constraints. Evaluate the objective function for total monthly profit at each of the five vertices of the graphed region. The vertices should occur at (0,0), (0,200),(300,200),(450,100),and (450,0).
The objective function is used to maximize or minimize a specific value, in this case, the total monthly profit. To formulate the objective function, we need to consider the profit for each type of television.
Let x represent the number of console televisions, and y represent the number of wide-screen televisions.
The profit from console televisions is $125 per unit, so the profit from selling x units of console televisions is 125x.
Similarly, the profit from wide-screen televisions is $200 per unit, so the profit from selling y units of wide-screen televisions is 200y.
Therefore, the objective function to maximize the total monthly profit can be written as:
P(x, y) = 125x + 200y
Now let's proceed to formulate the system of three inequalities that represents the constraints.
1. The first constraint is related to the production capacity. Let's assume the manufacturer can produce a maximum of 500 televisions per month.
x + y ≤ 500
This inequality ensures that the total number of console and wide-screen televisions produced does not exceed the manufacturing capacity.
2. The second constraint is that the manufacturer should produce at least 100 wide-screen televisions each month.
y ≥ 100
This inequality ensures that the minimum production requirement for wide-screen televisions is met.
3. The third constraint is that the manufacturer can only produce a maximum of 400 console televisions each month.
x ≤ 400
This inequality ensures that the maximum production limit for console televisions is not exceeded.
Now, let's evaluate the objective function at each of the five vertices of the graphed region:
Vertex 1: (0, 0)
P(0, 0) = 125(0) + 200(0) = $0
Vertex 2: (0, 200)
P(0, 200) = 125(0) + 200(200) = $40,000
Vertex 3: (300, 200)
P(300, 200) = 125(300) + 200(200) = $65,000
Vertex 4: (450, 100)
P(450, 100) = 125(450) + 200(100) = $70,500
Vertex 5: (450, 0)
P(450, 0) = 125(450) + 200(0) = $56,250
These values represent the monthly profit at each vertex of the graphed region, obtained by substituting the given values into the objective function.