(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:

Question

UpStudy AI · Accepted Answer

**Step 1: Define the Variables and the Objective Function**

Let  
$$
s = \text{number of software programs produced per week}
$$
and  
$$
v = \text{number of video games produced per week}
$$

The profit function we wish to maximize is:
$$
\text{Profit } P = 50s + 35v
$$

**Step 2: Write Down the Constraints**

The production constraints given by the problem are:
1. The company can produce at most 200 software programs:
   $$
   s \leq 200
   $$
2. The company can produce at most 300 video games:
   $$
   v \leq 300
   $$
3. The total production cannot exceed 425 items:
   $$
   s + v \leq 425
   $$
4. Non-negativity constraints:
   $$
   s \geq 0, \quad v \geq 0
   $$

**Step 3: Identify the Feasible Region and its Corner Points**

The feasible region is defined by the intersection of the constraints. The likely corner points are:

1. $$(s, v) = (0, 0)$$
2. $$(s, v) = (200, 0)$$ (using $$s = 200$$)
3. $$(s, v) = (0, 300)$$ (using $$v = 300$$)
4. Intersection of $$s = 200$$ and $$s + v = 425$$:
   $$
   v = 425 - 200 = 225 \quad \Rightarrow \quad (200, 225)
   $$
5. Intersection of $$v = 300$$ and $$s + v = 425$$:
   $$
   s = 425 - 300 = 125 \quad \Rightarrow \quad (125, 300)
   $$

**Step 4: Compute the Profit at Each Corner Point**

1. At $$(0, 0)$$:
   $$
   P = 50(0) + 35(0) = 0
   $$
2. At $$(200, 0)$$:
   $$
   P = 50(200) + 35(0) = 10000
   $$
3. At $$(0, 300)$$:
   $$
   P = 50(0) + 35(300) = 10500
   $$
4. At $$(200, 225)$$:
   $$
   P = 50(200) + 35(225) = 10000 + 7875 = 17875
   $$
5. At $$(125, 300)$$:
   $$
   P = 50(125) + 35(300) = 6250 + 10500 = 16750
   $$

**Step 5: Determine the Maximum Profit**

Comparing the computed profits, the maximum profit is 
$$
P = 17875
$$
which occurs at the corner point  
$$(s, v) = (200, 225)$$.

**Final Answer**

The company should produce $$200$$ software programs and $$225$$ video games per week to maximize the profit.
The company should produce 200 software programs and 225 video games per week to maximize the profit.

Upstudy AI Solution

Answer

Solution

Extra Insights

Related Questions

Latest Algebra Questions