Answer:
200 software program and 225 video games should be produced per week in order to maximize the profit
Step-by-step explanation:
Let us use the linear program to solve the problem
Assume that x represents the numbers of the software programs and y represents the numbers of the video games per week
∵ The company can produce at most 200 software programs
- At most means ≤
∴ x ≤ 200 ⇒ (1)
∵ The company can produce at most 300 video games
∴ y ≤ 300 ⇒ (2)
∵ The total production cannot exceed 425 items
- Can not exceed means ≤
∴ x + y ≤ 425 ⇒ (3)
Let us draw the graph of the three inequalities
Look to the attached graph
The red part represents x ≤ 200 ⇒ The shaded part is below the line x = 200
The blue part represents y ≤ 300 ⇒ The shaded pert is below the line y = 300
The green part represents x + y ≤ 425 ⇒ The shaded part is below the line x + y = 425
The common solution of the three inequalities is the polygon which contain the three colors together and its vertices are (0 , 0) , (200 , 0) , (200 , 225) , (125 , 300) , (0 , 300)
We will use this vertices to find the maximum profit
∵ The company makes a profit of $50 per software program
and $35 per video game
- Multiply x by 50 and y by 35, then add the products to get
the profit
∴ The profit = 50 x + 35 y
Substitute the coordinates of each vertex in the equation to find the maximum profit
At (200 , 0)
∵ x = 200 and y = 0
∴ The profit = 50(200) + 35(0)
∴ The profit = 10000 + 0
∴ The profit is $10,000
At (200 , 225)
∵ x = 200 and y = 225
∴ The profit = 50(200) + 35(225)
∴ The profit = 10000 + 7875
∴ The profit is $17,875
At (125 , 300)
∵ x = 125 and y = 300
∴ The profit = 50(125) + 35(300)
∴ The profit = 6250 + 10500
∴ The profit is $16,750
At (0 , 300)
∵ x = 0 and y = 300
∴ The profit = 50(0) + 35(300)
∴ The profit = 0 + 10500
∴ The profit is $10,500
The maximum profit is $17,875 with x = 200 and y = 225
200 software program and 225 video games should be produced per week in order to maximize the profit