100 points cartesian cola is a very popular drinks brand among mathematicians. they sell two products. a bottle of cartesian cola classic contains 500ml water, 600g sugar, and 0.1g caramel. a bottle of cartesian cola quantum contains 550ml water, 200g sugar, and 0.2g caramel. even though demand for their products is virtually unlimited, every day they are limited to 100 litres of water, 100kg of sugar and 32g caramel. for each bottle sold, cartesian cola classic makes 40p profit, and cartesian cola quantum makes 19p profit. how many bottles of each should cartesian cola produce each day to maximise their profit, and what is the profit per day?

Number of bottles of Cartesian Cola Classic produced each day: x>=0
Number of bottles of Cartesian Cola Quantum produced each day: y>=0
Limit of water each day: 100 litres = 100,000 ml
Limit of sugar each day: 100 kg = 100,000 g
Limit of caramel each day: 32 g

Quantity of water required each day:
500x+550y<=100,000.
Dividing the equation by 500:
500x/500+550y/500<=100,000/500
(1) x+1.1y<=200

Quantity of sugar required each day:
600x+200y<=100,000
Dividing the equation by 200:
600x/200+200y/200<=100,000/200
(2) 3x+y<=500

Quantity of caramel required each day:
0.1x+0.2y<=32
Dividing the equation by 0.1
0.1x/0.1+0.2y/0.1<=32/0.1
(3) x+2y<=320

Profit per day:
P=40x+19y

Please see the attached figure

The feasible region is formed by the vertices:
1) (0,0)
2) (0,160)
3) (160/3,400/3)
4) (3500/23,1000/23)
5) (500/3,0)

Evaluating the profit in each vertex:
1) (0,0)=(x,y)→x=0,y=0
P=40(0)+19(0)=0+0→P=0 p

2) (0,160)=(x,y)→x=0, y=160
P=40(0)+19(160)=0+3,040→P=3,040 p

3) (160/3,400/3)=(x,y)→x=160/3, y=400/3
P=40(160/3)+19(400/3)=6,400/3+7,600/3=(6,400+7,600)/3→
P=14,000/3→P=4,666.67 p

4) (3500/23,1000/23)=(x,y)→x=3500/23, y=1000/23
P=40(3500/23)+19(1000/23)=140,000/23+19,000/23
P=159,000/23→P=6,913.04 p

5) (500/3,0)=(x,y)→x=500/3, y=0
P=40(500/3)+19(0)=20,000/3+0
P=20,000/3→P=6,666.67 p

We get a maximum profit of 6,913.04 p when Cartesian Cola produces
152 (x=3,500/23=152.174) bottles of Cartesian Cola Classic and 43 (y=1,000/23=43.478) bottles of Cartesian Cola Quantum

Cartesian Cola should produce 152 bottles of Cartesian Cola Classic and 43 bottles of Cartesian Cola Quantum each day to maximise their profit.
The profit per day would be 6,913.04 p