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Question 48824: Hi, I need help solving this problem, I tried it, but I didn't get
far.
The dairy is capable of producing a maximum of 600 gallons of ice cream
each day.
Ingredients for regular brand ice cream cost $0.60 per gallon, and
premium brand costs $1.50 per gallon. The company can spend no more than
$675 per day on the ingredients.
Regular brand requires 4- person hours to produce 100 gallons of ice
cream, whereas premium brand requires 5- person hours to produce 100
gallons. The dairy is able to run 26 person hours per day of production.
Bo predicts that the dairy will make a profit of $2.88 per gallon on the regular brand and $3.24 per gallon on the premium.
Here's what I got: (p= premium, r= regular, P=profit)
Function that will determine daily profit for the dairy comparny:
Cost of ingredients per gallon produced each day:
Number of person hours to produce ice cream each day:
The question I'm having trouble with is: How many gallons of each brand
of ice cream should the diary produce to maximize it's daily profit,
taking into account all of the production requirements identified.
Provide supporting work that clearly describes your solution.
I believe the equation would be:
 --Is this correct, or are the hours have to be interpreted also?
I need help finding the amount of gallons of each brand that will make
the greatest profit. I created a graph and graphed the inequalities to find the feasible region and I know the amount of gallons of each should be in the feasible region, but how can I find the number of gallons for each???
Thanks much for your time.
Answer by venugopalramana(3286)   (Show Source):
You can put this solution on YOUR website! Hi, I need help solving this problem, I tried it, but I didn't get far.
The dairy is capable of producing a maximum of 600 gallons of ice cream each day.
Ingredients for regular brand ice cream cost $0.60 per gallon, and premium brand costs $1.50 per gallon. The company can spend no more than $675 per day on the ingredients.
Regular brand requires 4- person hours to produce 100 gallons of ice cream, whereas premium brand requires 4-(IS IT 5??) person hours to produce 100 gallons. The dairy is able to run 26 person hours per day of production.
Here's what I got: (p= premium, r= regular, P=profit)
Max number of gallons that can be produced each day:
p + r <= 600..................OK....p=600-r................I
Cost of ingredients per gallon produced each day:
1.5p + .6r <= 675.....................OK...p=(675-0.6r)/1.5..............II
Number of person hours to produce ice cream each day:
.04r + .05p <= 26..OK.IF 5 HRS ARE NEEDED FOR PREMIUM BRAND................p=(26-0.04r)/0.05.....III
The question I'm having trouble with is: How many gallons of each brand of ice cream should the diary produce to maximize it's daily profit, taking into account all of the production requirements identified. Provide supporting work that clearly describes your solution.
I believe the equation would be:
P= (.6r) + (1.5p) - 675 (Is this right?)............NO...PROFIT IS OBTAINED BY DEDUCTING THE COST PRICE FROM SALE PRICE.HERE THIS DATA IS NOT GIVEN.HENCE WE NEED THAT DATA SAY SALE PRICE OF THE 2 BRANDS TO KNOW THE PROFIT AND THEN MAXIMISE THE PROFIT . SUPPOSE IT IS MENTIONED THAT NET PROFIT ON SALE OF REGULAR BRAND IS SAY X AND NET PROFITON ON SALE OF PREMIUM BRAND IS Y THEN
P=rX+pY...WHICH WE WILL MAXIMISE SUBJECT TO THE 3 COSTRAINTS GIVEN BY EQNS.I,II
AND III.
P= (2.88r) + (3.24p)...............IV
NOW THAT THIS IS THERE WE CAN SOLVE.
WE HAVE 3 GIVEN CONSTRAINTS...VIDE EQNS.I,II,III ON r AND p AND 2 UNMENTIONED BUT OBVIOUS COSTRAINTS LIKE p>=0 AND r>=0...SUBJECT TO THESE WE TRY TO MAXIMISE EQN.IV.
WE DO HERE BY GRAPHICAL METHOD.PLOT ALL 3 EQNS,TAKING EQUALITY.
WE GOT 3 LINES REPRESENTING EQUALITY FOR THE 3 EQNS.BUT THE ZONE OF COMPLIANCE IS = OR < ...SO THE ZONE IS BELOW THE LINES TOWARDS ORIGIN.
NOW WE DRAW EQN.IV..GRAPH TAKING SLOPE=-2.88/3.24 WITH VARIABLE INTERCEPT ... SHIFTING IT TOWARDS THE PERMISSIBLE ZONE TO THE MAXIMUM EXTENT POSSIBLE AS THAT WILL GIVE US MAXIMUM PROFIT.THEORETICALLY THE MAXIMUM WILL OCCURR AT ONE OF THE INTERSECTION POINTS OF LINES I& II..OR...I & III...OR....II & III.HERE I HAVE DRAWN THE LINE THROUGH INTERSECTION OF LINES I & III AS SHIFTING THE LINE ANY FURTHER ILL TAKE IT BEYOND THE PERMISSIBLE ZONE AND HENCE THAT LINE GIVES US MAXIMUM PROFIT.IT EQUALS..THE POINT OF INTERSECTION IS r=400.....p=200
P=2.88*r+3.24*p=2.88*400+3.24*200=1800..YOU CAN CHECK THE OTHER POINTS OF INTERSECTION AND THE PROFIT THERE TO CONFIRM THAT THIS IS THE MAXIMUM PROFIT.THIS IS CALLED LINEAR PROGRAMMING AND YOU CAN HAVE FURTHER READING UNDER THAT HEAD.
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