Two fragrances A and B are used to make the perfumes Laughter and Joy.You require 3 g of fragrance A and 4 g of fragrance B to produce 1 litre of Laughter.One litre of Joy requires 9 g of fragrance A and 6 g of fragrance B.At least 3 litres of Laughter needs to be produced per week.At the beginning of a particular week the company has 27 g of fragrance A and 30 g of fragrance B. Let x and y be the number of litres of Laughter and Joy respectively that are produced per week.
Calculate the maximum possible profit.
We want to maximize the objective function P=30x+50y subject to the following constraints:
`3x+9ylt=27` (Every liter of Laughter requires 3g of A and every liter of Joy requires 9g of A -- the total A available is 27g.)
`4x+6y<=30` (Every liter of Laughter requires 4g of B, while every liter of Joy requires 6g of B -- the total B available is 30g.)
`xgt=3` (We are required to produce at least 3 liters of Laughter.)
And the natural constraints `x,y>=0` .
We graph the feasible region and locate the corner points.
The feasible region is closed so it has a maximum and a minimum. We substitute the coordinates of teh corner points into the objective function to find the maximum.
The corner points are (3,0),(7.5,0),(3,2) and (6,1).
The maximum is R230 and occurs when we manufacture 6L of Laughter and 1L of Joy.