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# Hot! Hot! Hot!

### Systems of Inequalities Mr. Kirkland April 8, 2014

Imagine that you are the owner of a hot sauce company that produces two different types of sauces, Red Hot Sauce and Scorchin' Hot Sauce.  Your goal would be to sell your product and maintain a profit to keep your business up and running.

This is the recipe that you have been using.  Each green pepper cost about \$1.00 while each chili pepper costs about \$0.50.  Now imagine that you bought 1050 green peppers and 1200 chili peppers.  What would be the most jars of sauce you could make with these?  You would need to create a system of inequalities to check it.

Your next step should be to graph these inequalities once you have simplified them.

The shaded area in the graph above is called the feasible region.  This means that you can take any point inside this region and plug it into you objective function and still make a profit.  Let's say that you make a \$1.20 profit on each pint of Red Hot Sauce and a \$1.00 profit on each pint of Scorchin' Hot Sauce. This is called your objection function.  But first, what you would want to do it take the coordinates of the corners and plug them into you objective function to find your min and max.

Next, you'll have orders to fill, right?  Well let's suggest that you can sell your sauce to a supermarket chain, a local grocery, and a specialty store. The supermarket chain will buy 288 pints every 8 weeks, the grocery will buy 60 pints every 4 weeks, and the speciality store will buy 24 pints each week.  How would you fill all these orders?  Well you would have to find the most number per week that you could make to fill all the orders on time.  In 8 weeks how many a week would you have to make to be done on time?  In 4 weeks how many? And 24 each week.  Now add all these together and that's how many a week you have to make.

Now to figure out how much you should sell each pint for you need yet another formula.

If each green pepper costs one dollar and each chili pepper costs 50 cents then you plug that into the recipe and for the Red Hot Sauce add \$1.20 to it and for Scorchin' Hot Sauce add one dollar to it for your profit.