National business machines manufactures x model a fax machines and y model b fax machines. each model a costs $100 to make, and each model b costs $150. the profits are $30 for each model a and $45 for each model b fax machine. if the total number of fax machines 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) =

Profit needs to be maximized.
Profit = 30x+45y where x and y are respectively the number of model A and model B fax machines manufactured.

Objective function:
max(30x+45y)

 Constraints:
x≥0 (1)
y≥0 (2)
x+y ≤ 2500 since the demand is capped at 2500 (3) 100x+150y≤600000 since manufacturing costs cannot exceed $600000(4)

 Solve the following two equations to identify where the two boundary lines (3) and (4) intersect.
x+y=2500(3)
100x+150y=600000---(4)

Multiplying (3) by 100
100x+100y=250000(5)
(5)-(4)
50y=350000
y=7000
x=-4500

since the constraint states that x≥0, only three vertices are considered viz (0,0), (0,2500),(2500,0).

applying the profit function at each of the three vertices:
(0,0) 30(0)+45(0) = 0
(0,2500) 30(0)+45(2500)=112500
(2500,0) 30(2500)+45(0)=75000

Hence by applying the max function, x=0, y=2500.
i.e. Dont produce any 'a' model machine. Manufacture 2500 units of model 'b' to maximise profit

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