Need help with this work problem please?

[cantfigurethisout]

Your company makes two different kinds of sauce, Red Hot Sauce and Scorchin' Hot Sauce. As the owner of a small but successful business, you want to minimize costs, maximize profit, and create satisfied customers by filling orders promptly.


As you work through the project, you will use systems of equations and a spreadsheet to analyze production levels and make decisions. You will write a report detailing your choices.


Activity 1: Graphing
To fill an order for Sizzlin' Sauce sauces, you bought 1050 green peppers and 1200 hot chili peppers.


Write and graph a system of inequalities to represent how many pints of each kind of sauce you can make. Use the recipes below.
Select one solution of the system and determine how many peppers you will have left over.


Sizzlin' Sauces Recipes:
Scorchin' Hot Sauce Ingredients
Yield: 1 pint
1 pint tomato sauce with onions
4 green peppers, diced
8 hot chili peppers, seeded and diced


Red Hot Sauce Ingredients
Yield: 1 pint
1 pint tomato sauce with onions
5 green peppers, diced
4 hot chili peppers, seeded and diced


Activity 2: Analyzing
Suppose you make $1.20/pt profit on Red Hot Sauce and $1.00/pt profit on Scorchin' Hot Sauce. Using the restrictions from Activity 1, decide how much of each sauce you should make and sell to maximize your profit. What is the maximum profit?


Activity 3: Researching
Visit a local grocery store to estimate the cost of each sauce ingredient. Remember that buying in large quantities can save you money.


Find the cost to produce 1 pt of each type of sauce.
What selling price will you set for each sauce to maintain your profit?


Activity 4: Organizing
You can sell your sauce to a supermarket chain, a local grocery store, and a specialty store. The supermarket chain will buy 288 pints at a time, every eight weeks. The grocery store will buy 60 pints every four weeks, and the specialty store will buy 24 pints each week. How much sauce should you produce each week to fill these orders? Presume that you want to produce the same number of pints each week, and that the type of sauce is not a factor in filling these orders.
Design a spreadsheet to track your stock of sauce after each week. Use cell formulas.

 

[cantfigurethisout]

This is what I have so far

So far i have x=#of bottles of red sauce, y=# of bottles of scorchin sauce.
I'm not 100% sure but I think the equations for activity 1 are 5x+4y<=1050, and 4x+8y<=1200? And then the two graphs intersect at x=150, y=75. Where do I go from here..? Am I done with activity 1? How do I do activity 2?

 

[JeffM]

So far i have x=#of bottles of red sauce, y=# of bottles of scorchin sauce. Good first step: name your variables. I'm not 100% sure but I think the inequations for activity 1 are 5x+4y<=1050, and 4x+8y<=1200? Correct but incomplete. And then the two graphs intersect at x=150, y=75. The equations that bound the identified inequalities do intersect at x = 150 and y = 75, but it may not be relevant. Where do I go from here..? Am I done with activity 1? You are not. How do I do activity 2?

Please read the the thread entitled "Read Before Posting."

Are you studying linear programming? Whether you are or not, there are four relevant constraining inequalities and two extra constraints.

\(\displaystyle [\alpha]\ 0 \le x.\) You can hardly make a negative number of pints. This is a non-negativity constraint.

\(\displaystyle [\beta]\ 0 \le y.\) Another non-negativity constraint.

\(\displaystyle [\gamma]\ 4x + 8y \le 1050.\) A positive constraint.

\(\displaystyle [\delta]\ 5x + 4y \le 1200.\) Another positive constraint.

\(\displaystyle [\epsilon]\ x \in \mathbb Z.\) x must be an integer. Your teacher may ignore this constrain as well as the next one.

\(\displaystyle [\zeta]\ y \in \mathbb Z.\)

The four straight lines define an area, what I was taught to call the feasible area. Any point with integer ordinates within this area identifies a combination of the two sauces that it is feasible to produce. Any point outside this area represents a combination of the two sauces that it is not feasible to produce. Do you see why? (Again, your teacher may ignore the limitation to integers.)

In activity 1, you are asked to pick a point in the feasible region and determine from that how many peppers of each type are not used. To make sure you understand, I'd pick several point inside and outside the feasible region and pick x = 150 and y = 75.

See how far all that gets you? It should get you to understand fully what the feasible area means in practical terms

 

[cantfigurethisout]

Ok now I'm getting somewhere

Ok thank you! So that helps settle my questions for activity 1 & 2. Now my next question is to how I start Activity 4?

"You can sell your sauce to a supermarket chain, a local grocery store, and a specialty store. The supermarket chain will buy 288 pints at a time, every eight weeks. The grocery store will buy 60 pints every four weeks, and the specialty store will buy 24 pints each week.
How much sauce should you produce each week to fill these orders? Presume that you want to produce the same number of pints each week, and that the type of sauce is not a factor in filling these orders. Design a spreadsheet to track your stock of sauce after each week. Use cell formulas."

So now do I need to make an inequality graph for the different stores? (for example x=# of weeks it takes to buy product, y=# of pints they purchase)

 

[JeffM]

Ok thank you! So that helps settle my questions for activity 1 & 2. Now my next question is to how I start Activity 4?

"You can sell your sauce to a supermarket chain, a local grocery store, and a specialty store. The supermarket chain will buy 288 pints at a time, every eight weeks. The grocery store will buy 60 pints every four weeks, and the specialty store will buy 24 pints each week.
How much sauce should you produce each week to fill these orders? Presume that you want to produce the same number of pints each week, and that the type of sauce is not a factor in filling these orders. Design a spreadsheet to track your stock of sauce after each week. Use cell formulas."

So now do I need to make an inequality graph for the different stores? (for example x=# of weeks it takes to buy product, y=# of pints they purchase)

The problem as stated is somewhat ambiguous because you are not told when the orders must be filled. Let's assume that you must fill all three orders on Monday of week1. So what is the minimum that you must have in inventory at the end of week 0? What is the amount you must produce each week? You now have all the information required to set up your spread sheet.