Answer: The maximum profit is $19,000, and to maximize profit, the company should produce 200 software programs and 200 video games per week.
Step-by-step explanation:
a) The function we are trying to maximize is the profit. Let's define the variables:
- x: number of software programs produced per week
- y: number of video games produced per week
The profit function can be written as:
P(x, y) = 40x + 55y
b) Now, let's write the system of inequalities that describes the constraints:
- The company can produce at most 200 software programs per week, so x ≤ 200.
- The company can produce at most 300 video games per week, so y ≤ 300.
- Total production cannot exceed 400 items per week, so x + y ≤ 400.
c) To graph the system of inequalities, we need to plot the lines corresponding to the three constraints.
The line for x ≤ 200 is a vertical line passing through the point (200, 0).
The line for y ≤ 300 is a horizontal line passing through the point (0, 300).
The line for x + y ≤ 400 is a diagonal line with a slope of -1, passing through the points (0, 400) and (400, 0).
The graph of the system of inequalities would be a triangle in the first quadrant, with vertices at (0, 0), (0, 300), and (200, 200).
d) To find the maximum profit, we need to evaluate the profit function at each vertex of the feasible region and compare the results.
At the vertex (0, 0), the profit is P(0, 0) = 40(0) + 55(0) = $0.
At the vertex (0, 300), the profit is P(0, 300) = 40(0) + 55(300) = $16,500.
At the vertex (200, 200), the profit is P(200, 200) = 40(200) + 55(200) = $19,000.
Therefore, the maximum profit is $19,000, and to maximize profit, the company should produce 200 software programs and 200 video games per week.