A small firm manufactures and sells litre cartons of non-alcoholic cocktails, ‘The Caribbean’ and ‘Mr.Fruity’, which sell for $1 and $1.25, respectively. Each is made by mixing fresh orange, pineapple and apple juices in different proportions. The Caribbean consists of 1 part orange, 6 parts pineapple and 1 part apple. Mr Fruity consists of 2 parts orange, 3 parts pineapple and 1 part apple. The firm can buy up to 300 litres of orange juice, up to 1125 litres of pineapple juice and up to 195 litres of apple juice each week at a cost of $0.72, $0.64 and $0.48 per litre, respectively.
Find the number of cartons of ‘The Caribbean’ and ‘Mr Fruity’ that the firm should produce to maximize profits. You may assume that non-alcoholic cocktails are so popular that the firm can sell all that it produces.
Find the number of cartons of ‘The Caribbean’ and ‘Mr Fruity’ that the firm should produce to maximize profits. You may assume that non-alcoholic cocktails are so popular that the firm can sell all that it produces.
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Let x = no. of liter cartons of ‘The Caribbean’ and let y = no. liter of cartons of ‘Mr Fruity’
The Caribbean consists of 1 part orange, 6 parts pineapple and 1 part apple.
(1+6+1=8)
==> 1 liter of The Caribbean is 1/8 orange, 6/8=3/4 pineapple and 1/8 apple.
==> x liters of The Caribbean is x/8 orange, 3x/4 pineapple and x/8 apple.
Mr Fruity consists of 2 parts orange, 3 parts pineapple and 1 part apple.
(2+3+1=6)
==> 1 liter of Mr Fruity is 2/6=1/3 orange, 3/6=1/2 pineapple and 1/6 apple.
==> y liters of Mr Fruity is y/3 orange, y/2 pineapple and y/6 apple.
Total litres of orange juice: x/8 + y/3
x/8 + y/3 <=300
Total litres of pineapple juice: 3x/4 + y/2
3x/4 + y/2 <=1125
Total litres of apple juice: x/8 + y/6
x/8 + y/6 <=195
Let z = profit.
Profit = Revenue - Cost
Revenue = 1x + 1.25y
Cost = 0.72(x/8 + y/3) + 0.64(3x/4 + y/2) + 0.48(x/8 + y/6) = 0.63 x+0.64 y
==> z = profit = (1x + 1.25y) - (0.63 x+0.64 y) = 0.37x + 0.61y
Maximize z = 0.37x + 0.61y
subject to the ff constraints:
x/8 + y/3 <=300
3x/4 + y/2 <=1125
x/8 + y/6 <=195
x,y>=0
The region bounded by the constraints are graphed here:
http://www.wolframalpha.com/input/?i=x%2…
Determine the vertices of the region. Plug in the values of x and y into the formula for z and determine which ordered pair yields the largest z.
The Caribbean consists of 1 part orange, 6 parts pineapple and 1 part apple.
(1+6+1=8)
==> 1 liter of The Caribbean is 1/8 orange, 6/8=3/4 pineapple and 1/8 apple.
==> x liters of The Caribbean is x/8 orange, 3x/4 pineapple and x/8 apple.
Mr Fruity consists of 2 parts orange, 3 parts pineapple and 1 part apple.
(2+3+1=6)
==> 1 liter of Mr Fruity is 2/6=1/3 orange, 3/6=1/2 pineapple and 1/6 apple.
==> y liters of Mr Fruity is y/3 orange, y/2 pineapple and y/6 apple.
Total litres of orange juice: x/8 + y/3
x/8 + y/3 <=300
Total litres of pineapple juice: 3x/4 + y/2
3x/4 + y/2 <=1125
Total litres of apple juice: x/8 + y/6
x/8 + y/6 <=195
Let z = profit.
Profit = Revenue - Cost
Revenue = 1x + 1.25y
Cost = 0.72(x/8 + y/3) + 0.64(3x/4 + y/2) + 0.48(x/8 + y/6) = 0.63 x+0.64 y
==> z = profit = (1x + 1.25y) - (0.63 x+0.64 y) = 0.37x + 0.61y
Maximize z = 0.37x + 0.61y
subject to the ff constraints:
x/8 + y/3 <=300
3x/4 + y/2 <=1125
x/8 + y/6 <=195
x,y>=0
The region bounded by the constraints are graphed here:
http://www.wolframalpha.com/input/?i=x%2…
Determine the vertices of the region. Plug in the values of x and y into the formula for z and determine which ordered pair yields the largest z.