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.

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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).

P=30x+50y:

P(3,0)=90

P(7.5,0)=225

P(3,2)=190

P(6,1)=230

**The maximum is R230 and occurs when we manufacture 6L of Laughter and 1L of Joy.**

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