# Math Problem Help



## CorwinShiu (May 5, 2008)

Hey I need help on a word problem. How would you set this up?

"A juice company makes two kinds of juice: Orangeade and Berry-fruity. One gallon of Orangeade is made by mixing 2.5 quarts of orange juice and 1.5 quarts of raspberry juice, while one gallon of Berry-fruity is made by mixing 3 quarts of raspberry juice and 1 quart of orange juice. A profit of $.50 is made on every gallon of Orangeade sold, and a profit of $.40 is made on every gallon of Berry-fruity sold. If the company has 150 gallons of raspberry juice and 125 gallons of orange juice on hand, how many gallons of each type of juice should be made to maximize profit?"

The Objective function is P = .5x + .4y where x is orangeade, y is berry-fruity and P is profit.

The constrait is the amounts of orange juice and raspberry juice but I don't know how to set up the equations. Can anyone explain? Thanks.


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## Dene (May 5, 2008)

I think the best way is to write an equation so one variable is in terms of the other (I can't be bothered to work it out, but something like 5x=2y-1 or whatever). Then substitute the new equation into the old one and play around with it a bit.


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## cmhardw (May 5, 2008)

Break down each of orangeade and Berry-fruity into their component parts.

For example, their are x gallons of orangeade. Orange juice is 5/8 of Orangeade and Raspberry is 3/8 based on your proportion.

There are y gallons of Berry-fruity. Orange juice is 1/4 of this and Raspberry is 3/4 of this.

Now add up all the orange juice used from both drinks:
x*(5/8) + y*(1/4) <= 125

This says that the amount of orange juice in Orangeade plus the amount of orange juice in Berry fruity must be less than or equal to the total available 125 gallons.

Do the same for raspberry to get:
x*(3/8)+y*(3/4) <= 150

Graph this on a graph where the x axis is the amount of orangeade and the y axis is the amount of berry-fruity. Your maximal point for the profit function must occur at a corner of the solution region. Test the four corner points and see which one gives you the most profit.

---edit---
I think it should be implicitly understood that x >= 0 and y >= 0 as well
---edit---

Hope this helps,
Chris


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