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.
Please show work.

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

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