let y = the number of liters of mister fruity.
one carton of either drink is equal to 1 liter.
the caribbean consists of 1 part orange juice, 6 parts pineapple juice, and 1 part apple juice.
mister fruity consists of 2 parts orange juice, 3 parts pineapple juice, and 1 part apple juice.
total number of parts for the caribbean is 8.
total number of parts for mister fruity is 6.
since the total number of liters of each drink is 1, then:
each part of the caribbean is 1/8 of 1 liter.
each part of mister fruity is 1/6 of 1 liter.
since the caribbean consists of 1 part orange juice, 6 parts pineapple juice, and 1 part apple juice, then the caribbean uses 1/8 of a liter of orange juice, 6/8 of a liter of pineapple juice, 1/8 of a liter of apple juice.
since mister fruity consists of 2 parts orange juice, 3 parts pineapple juice, and 1 part apple juice, then mister fruity uses 2/6 of a liter of orange juice, 3/6 of a liter of pineapple juice, 1/6 of a liter of apple juice.
the firm can buy up to: 300 liters of orange juice, 1125 liters of pineapple juice, 195 liters of apple juice.
since the caribbean uses 1/8 of a liter of orange juice and mister fruity uses 2/6 of a liter of orange juice, then your constraint equation for orange juice is:
1/8 * x + 2/6 * y <= 300
since the caribbean uses 6/8 of a liter of pineapple juice and mister fruity uses 3/6 of a liter of pineapple juice, then your constraint equation for pineapple is:
6/8 * x + 3/6 * y <= 1125
since the caribbean uses 1/8 of a liter of apple juice and mister fruity uses 1/6 of a liter of apple juice, then your constraint equation for apple juice is:
1/8 * x + 1/6 * y <= 195
another constraint is that both x and y must be greater than or equal to 0.
the cost for orange juice is .74 per liter.
the cost for pineapple juice is .64 per liter.
the cost for apple juice is .48 per liter.
the selling price of the caribbean is 1 dollar per liter.
the selling price of mister fruity is 1.25 dollars per liter.
the profit equation is the selling price minus the cost.
for the caribbean, the profit equation will be:
1.00 - 1/8 * .74 - 6/8 * .64 - 1/8 * .48 = .3675.
for mister fruity, the profit equation will be:
1.25 - 2/6 * .74 - 3/6 * .64 - 1/6 * .48 = .6033333333.
the profit equation is your objective function and is shown below:
profit = .3675 * x + .6033333333 * y
the constraint functions are:
1/8 * x + 2/6 * y <= 300 (orange juice)
6/8 * x + 3/6 * y <= 1125 (pineapple juice)
1/8 * x + 1/6 * y <= 195 (apple juice)
x >= 0
y >= 0
using the desmos.com calculator, you will graph the opposite of the inequalities.
the area of the graph that is not shaded is your region of feasibility.
the corner points of the region of feasibility will contain the maximum profit.
each of those corner points will be evaluated by the objective function to determine which of them contains the maximum profit.
the coordinate points are in (x,y) format.
x represents the number of liters of the caribbean.
y represents the number of liters of mister fruity.
the graph looks like this:
all the constraints must be satisfied.
x >= 0 and y >= 0 is satisfied.
1/8 * x + 2/6 * y <= 300 (orange juice) is satisfied because 1/8 * 720 + 2/6 * 630 = 300 which is equal to 300.
6/8 * x + 3/6 * y <= 1125 (pineapple juice) is satisfied because 720 * 6/8 + 3/6 * 630 = 855 which is less than 1125.
1/8 * x + 1/6 * y <= 195 (apple juice) is satisfied because 720 * 1/8 + 630 * 1/6 = 168.75 which is less than 195.
if i did this right (always a consideration), your solution is that 720 cartons of the carribean and 630 cartons of mister fruity should be produced and sold to generate the maqximum profit.