Help with Algebra II? (Systems of Inequalities/Linear Programming)

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prestico23

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Okay so basically I have to do this chapter project for my algebra II class and I've gotten completely stuck.

Suppose you are the owner of the Sizzlin' Sauce Company. 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.

List of Materials



  • Calculator
  • Graph paper

Activities


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





I've set up Activity 1 so that the inequalities are 5x + 4y <= 1050 and 4x + 8y <=1200. Then I graphed the two equations but now I'm completely stuck on the Activity 2 part of the project. What restrictions does it mean and how do I figure out the best way to approach the problem?
 
Not sure what you mean by definitions...sorry i'm terrible at algebra. I used X for the green peppers and Y for the hot chili peppers if that is what you meant.
 
prestico23 said:
Not sure what you mean by definitions
Click to expand...

In algebra, the symbols in a word problem represent quantities.

In other words, x is some specific quantity.

To define symbol x means to state the specific quantity that symbol x represents.



I used X for the green peppers and Y for the hot chili peppers
Click to expand...

First, a note about notation. In algebra, symbols x and X represent two different things. It's good form to be consistent. If you start out writing x and y, continue using those symbols; upper- and lower-case letters are not interchangeable, in algebra.

So, it seems like you're thinking about the following definitions.

x = the number of green peppers used by the Sizzlin' Sauce Company

y = the number of hot chili peppers used by the Sizzlin' Sauce Company

That's not correct. Look what your inequality says, using these definitions:

5x + 4y <= 1050

"Five times the number of green peppers used by the company plus four times the number of chili peppers used by the company must be less than or equal to 1,050 peppers"

That doesn't make sense, does it?

Try to define the variables in terms of the number of bottles of sauce produced, instead.

Also, your system of inequalities must include statements about whether x and y must be non-negative numbers. In other words, you should have four inequalities, in your system.
 
Well in the first equation I meant it so that it would be "Five times the number of green peppers used by the company (Scorchin Sauce) plus four times the number of green peppers used by the company (Red Sauce) must be less than or equal to 1,050 green peppers" (Total). I may have wrote it incorrectly. Even still, I may have to re-adjust my inequalities so that the inequalities describe the number of bottles produced instead but I'm not sure how to go about doing that
 
Your inequalities will work, but not with your current definitions for symbols x and y.

Here is how you write the definitions:

x = the number of bottles of Red Hot Sauce produced

y = the number of bottles of Scorchin' Hot Sauce produced

With these definitions, the expression 5x represents the total number of green peppers used in the production of x bottles of Red Hot Sauce.

Likewise, the expression 4y represents the total number of green peppers used in the production of y bottles of Scorchin' Hot Sauce.

Therefore, the expression 5x + 4y represents the total number of green peppers used by the company for everything.

Make sense, now?
 
Yes, thank you, I appreciate you clarifying that for me. Now I'm just not sure where to begin or what to do exactly for activity 2.
 
We are not ready for activity 2, yet. :cool:

You need four inequalities; so far, you've only written two.

What can you say about the numbers x and y. Can these symbols represent any Real number?
 
Fair enough :). Well if x and y represent the number of bottles produced then they both have to be above 0. So I'm assuming you'd need to write x>=0 and y>=0. Would those be the other two inequalities used?
 
prestico23 said:
Well if x and y represent the number of bottles produced then they both have to be above 0.

x>=0

y>=0


Would those be the other two inequalities used?
Click to expand...

Close, but not quite.

Compare your blue statement with your red statements.
 
Very good. Now you have a system of four inequalities to graph.

x > 0

y > 0

5x + 4y <= 1050

4x + 8y <= 1200


Technically speaking, such a graph would include shading four overlapping regions; the area of overlap forms some sort of irregular polygon.

The ultimate solution about profit will appear at a vertex of this shape; in other words, the x- and y- coordinates at the vertices give the numbers to check in what's called the "objective function". One of those pairs of x and y values comprise the profit solution. But, we're not ready to talk about the objective function or profit, yet.

(Writing and testing vertex coordinates in the objective function is what we do in activity 2. Right now, you need to draw the graph. That is, you need to graph the system of inequalities, in order to discover the solution set.)


Don't be concerned about the shading, on your graph. Although, you should understand how such shading forms the shape of the irregular polygon (i.e., the solution set), even if you don't shade.


You need to graph the following lines:

x = 0 (graphs as dotted line)

y = 0 (graphs as dotted line)

5x + 4y = 1050 (graphs as solid line)

4x + 8y = 1200 (graphs as solid line)


Can you do this?

If so, please state the vertex coordinates at the four "corners" of the resulting region.
 
Last edited:
I attempted to graph the 4 inequalities but i'm relatively sure I did something wrong. When i graphed the 4 lines, it created what looks like a right triangle with the vertices at (0,0), (0, 150) and (6,0). I'm not entirely sure what I did wrong. I think graphing with such large numbers threw me off. I turned the first two inequalities into SI form, y<= 5/4x + 262.5 and y<= -.5x + 150. x>0 and y>0 are pretty straightforward but how should i go about graphing those?
 
prestico23 said:
it created what looks like a right triangle with the vertices at (0,0), (0, 150) and (6,0).

y<= 5/4x + 262.5

y<= -.5x + 150

x>0 and y>0 are pretty straightforward but how should i go about graphing those?
Click to expand...

We're graphing lines, yes?

x = 0

y = 0

If you need help with these two, let me know.

y = -1/2 + 150 is correct

y = 5/4*x + 262.5 is not correct; double-check your work converting to slope-intercept form


EDIT: Please ignore my previous (deleted) statement about the system's solution set forming a three-sided region. That statement was incorrect; I made an error coding the computer commands.
 
Last edited:
Are your vertex coordinates (6, 0) another typographical error?

You should have a total of four vertices.
 
Okay. Please try again.

If your second attempt does not yield the correct four vertex coordinates, I will post the graph.

(0,0) is correct

(0,150) is correct

There is a vertex on the positive x-axis, and there is a vertex in Quadrant I.
 
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