What is the optimal profit?

Question

National Business Machines manufactures x model A portable printers and y model B portable printers. Each model A costs $100 to make, and each model B costs $150. The profits are $45 for each model A and $35 for each model B portable printer. If the total number of portable printers demanded per month does not exceed 2500 and the company has earmarked no more than $600,000/month for manufacturing costs, how many units of each model should National make each month to maximize its monthly profit?

 

(x,y)= (   ?    ,    ?    )

 

Optimal Profit               ?

Answer 1

Hi Brittany,

 

Let's begin by assigning letters to represent the number of production units for each type of portable printer:

    x = # of "A" type portable printers

    y = # of "B" type portable printers

 

Next, let's write the system of linear inequalities to reflect our given constraints:

    1.  Cost Constraint (100x + 350y ≤ 600,000)

    2.  Production Constraint (x + y ≤ 2500)

 

Next, let's write these inequalities as a system of linear equations to be solved for their common point of intersection, which will represent our optimal production mix for these given constraints:

    Equation #1:  100x + 350y = 600,000

    Equation #2:  x + y = 2500

 

Next,  let's multiply each term in Equation #2 and then solve this system of linear equations via the Elimination technique:

    100x + 350y = 600,000

 - (100x + 100y = 250,000)

 --------------------------------

               250y = 350,000

                    y = 350,000/250

                    y = 1400 "B" type printers

              x + y = 2500 (Equation #2)

              x + 1400 = 2500

                    x = 2500 - 1400

                    x = 1100 "A" type printers

 

(x,y) = (1100,1400)

 

Finally, we'll calculate the optimal profit by plugging in our optimal values (x,y) into our profit equation:

    Profit = 45x + 35y

    Profit = 45(1100) + 35(1400)

    Profit = 43,500 + 49,000

    Profit = 92,500

 

Optimal Profit = $92,500

 

Below is the link to our graph for the given constraints showing their intersection point (1100,1400), which confirms our algebraic solution above:

 

https://dl.dropbox.com/s/dlwfamqei5wfy75/Graph_of_Optimal_Profit_for_Production_Mix_of_Portable_Printers.png?raw=1

 

Thanks for submitting this problem and glad to help.

 

God bless, Jordan.