(Score for Question 3:__of 5 points) 3. A company makes a profit of \( \$ 50 \) per software program and \( \$ 35 \) per video game. The company can produce at most 200 software programs and at most 300 video games per week. Total production cannot exceed 425 items per week. How many items of each kind should be produced per week in order to maximize the profit? Use linear programming to solve. Show all your work. Answer:

To solve this linear programming problem, we need to define our variables, formulate the objective function, establish the constraints, and then graph or calculate the optimal solution. **Step 1: Define the Variables** Let: - \( x \) = number of software programs produced - \( y \) = number of video games produced **Step 2: Objective Function** We want to maximize profit: \[ P = 50x + 35y \] **Step 3: Constraints** From the problem, we have the following constraints: 1. \( x \leq 200 \) (maximum software programs) 2. \( y \leq 300 \) (maximum video games) 3. \( x + y \leq 425 \) (total production limit) 4. \( x \geq 0 \) (cannot produce negative items) 5. \( y \geq 0 \) **Step 4: Graphing the Constraints** To visualize the constraints on a graph: - For \( x = 200 \), draw a vertical line at \( x = 200 \). - For \( y = 300 \), draw a horizontal line at \( y = 300 \). - For \( x + y = 425 \), convert this to slope-intercept form: \( y = -x + 425 \). This line crosses the x-axis at 425 and the y-axis at 425. **Step 5: Identify the Feasible Region** The feasible region is the area where all the constraints overlap. This is bounded by the lines we've just illustrated. **Step 6: Determine the Corner Points** The corner points of the feasible region can be identified from the intersections of the constraint lines: 1. (200, 225): intersection of \( x = 200 \) and \( x + y = 425 \) 2. (125, 300): intersection of \( y = 300 \) and \( x + y = 425 \) 3. (0, 300): intersection of \( y = 300 \) and \( x = 0 \) 4. (200, 0): intersection of \( x = 200 \) and \( y = 0 \) 5. (0, 425): intersection of \( x + y = 425 \) and \( x = 0 \) **Step 7: Evaluate the Objective Function at Each Corner Point** 1. At (200, 225): \( P = 50(200) + 35(225) = 10000 + 7875 = 17875 \) 2. At (125, 300): \( P = 50(125) + 35(300) = 6250 + 10500 = 16750 \) 3. At (0, 300): \( P = 50(0) + 35(300) = 0 + 10500 = 10500 \) 4. At (200, 0): \( P = 50(200) + 35(0) = 10000 + 0 = 10000 \) 5. At (0, 425): not applicable since this point does not satisfy other maximum constraints. **Step 8: Determine the Optimal Solution** The maximum profit occurs at the corner point (200, 225) where: - produce 200 software programs - produce 225 video games Thus, the company should produce **200 software programs and 225 video games** per week to maximize the profit, yielding a maximum profit of \( \$17,875 \).