Nutty Economics!

 

Introduction

Linear systems of equations and inequalities are used widely in the fields of business and economics to maximize profit, minimize cost, and balance the production and consumption of goods.  In fact, there are entire careers in Production and Operations Management that are built around this skill.

 

Task

In this project you will play the role of consultant to a retailer of nuts.  You will apply your knowledge of systems of linear equations and inequalities to balance the production of two types of goods in such a way that the profit will be maximized.

 

Instructions

Work with at least one other person to complete the following exercises.  Solve each problem in order and save your work as you progress as you will create a professional presentation in the end.

 

·         First Problem – Defining the Quantities:  The Golden Nut specialty store has obtained 60 lbs. of cashews and 50 lbs. of premium almonds at a good price from its supplier.  Since the almonds are more expensive and more of a delicacy than the cashews, it is better to sell the cashews mixed with almonds in order to get a better price for them.  The storeowner decides to sell two products: Hearty Nut Mix and Premium Blend.  The company hires you to determine how much of each product they should mix together to maximize their profit.

 

o        If each pound of Hearty Nut Mix contains 4 ounces of almonds and 12 ounces of cashews, write an expression that represents the number of pounds of each nut in x pounds of the mix.  [Hint: What fraction of the pound is cashews, and what fraction is almonds?]

 

o        If each pound of Premium Blend contains 8 ounces of almonds and 8 ounces of cashews, write an expression that represents the number of pounds of each nut in y pounds of the mix.

 

o        Complete the following table and use it to write an expression that represents the total number of pounds of cashews used for both mixes together.  Then, write an expression for the total number of almonds used for both mixes together.

 

Type of Mix

No. of Pounds of Mix

No. of Pounds of Cashews

No. of Pounds of Almonds

Hearty Nut Mix

x

 

 

Premium Blend

y

 

 

 

o        Finally, your market research shows that since the Premium Blend is much more popular, you can afford to charge quite a bit more for it, obtaining a profit of $0.45 per pound for it while you can only earn a profit of $0.30 per pound for the Hearty Nut Mix.  Write an expression for the amount of profit that you would make from selling x pounds of Hearty Nut Mix and y pounds of Premium Blend.

 

·         Second Problem – Constraints:  It might seem like selling all Premium Blend, since it obtains a better profit, might give the most profit.  But this is not necessarily so.  Explain why.  [Hint: Remember that we are not going to sell cashews or almonds by themselves at this point.]

 

o        You have a limited amount of each type of nut.  This places a constraint on the problem that we must account for.  Write an inequality that expresses the fact that the total number of cashews cannot be greater than 60 lbs. and the total number of almonds cannot be greater than 50 lbs.  [Hint: Use the table from the first problem above.]

 

o        Write down two other inequalities that represent constraints on the variables x and y.  [Hint: What types of values for x and y make sense considering that they represent pounds?  These will be simple inequalities.]

 

·         Third Problem – Possibilities:  The next step in the process is to graph the inequalities and determine the points that satisfy all of them at once, because these are the only values of x and y that we can realistically use with our supply of nuts.  Sketch all four inequalities on the same graph and shade in the region that represents those points that satisfy all of the inequalities.

 

·         Fourth Problem – Maximizing Profit:  We are now ready to determine the maximum profit.

 

o        The points in the region you graphed represent all possible x and y values that we would be allowed to use to maximize our profit.  This is a lot of points to check just to find the maximum profit!  Fortunately, we can eliminate many of them.  It turns out that both the maximum and minimum profit will occur on the edges of this region (in fact, on the extreme points of the edge—the corners).  Therefore, compute the x and y coordinates of each of the corners of your region.  This will require you to solve some systems of equations.

 

o        Since we know the maximum (and minimum) profit must occur at one of these corners, plug the x and y values for each corner into the expression for profit from the first problem, and record your answers in the table below.  Then, identify the maximum profit and tell how many pounds of each mix should be made.  [Note: The table below may contain more rows than you need to complete the problem.]

 

 

Expression for Profit:_________________________________

 

Corner (x,y)

Profit

 

 

 

 

 

 

 

 

 

 

 

 

 

 

o        By completing the following chart, determine how many nuts are left unmixed.

 

Mix

Pounds Made

Pounds of Cashews in the Mix

Pounds of Almonds in the Mix

Hearty Nut Mix

 

 

 

Premium Blend

 

 

 

 

 

Total Pounds of Cashews Used=

 

 

Total Pounds of Almonds Used=

 

 

 

 

Conclusions

Compare your expressions, computations, and the graph with those from another group.  Work to make sure that your explanation is clear and concise. 

 

Prepare a final report that clearly identifies and explains

·         the quantities involved in the problem,

·         the expression for profit,

·         the constraints,

·         the region of possible x and y values,

·         the possible combination of mixes that could result in maximum profit, and

·         the maximum profit and the number of pounds of each mix that should be made.

 

Finally, present your solution to the class.

Grade

Your project should correctly identify and justify mathematically: (1) the quantities involved in the problem, (2) the expression for profit, (3) the constraints, (4) the region of possible x and y values, (5) the possible combination of mixes that could result in maximum profit, and (6) the maximum profit and the number of pounds of each mix that should be made.

 

 

Score

Content

Presentation/Communication

4

•       The solution shows a deep understanding of the problem including the ability to identify the appropriate mathematical concepts and the information necessary for its solution.

•       The solution completely addresses all mathematical components presented in the task.

•       The solution puts to use the underlying mathematical concepts upon which the task is designed and applies procedures accurately to correctly solve the problem and verify the results.

•       Mathematically relevant observations and/or connections are made.

•       There is a clear, effective explanation detailing how the problem is solved. All of the steps are included so that the reader does not need to infer how and why decisions were made.

•       Mathematical representation is actively used as a means of communicating ideas related to the solution of the problem.

•       There is precise and appropriate use of mathematical terminology and notation.

•       Your project is professional looking with graphics and effective use of color. 

3

•       The solution shows that the student has a broad understanding of the problem and the major concepts necessary for its solution.

•       The solution addresses all of the mathematical components presented in the task.

•       The student uses a strategy that includes mathematical procedures and some mathematical reasoning that leads to a solution of the problem.

•       Most parts of the project are correct with only minor mathematical errors.

•       There is a clear explanation.

•       There is appropriate use of accurate mathematical representation.

•       There is effective use of mathematical terminology and notation.

•       Your project is neat with graphics and effective use of color. 

2

•       The solution is not complete indicating that parts of the problem are not understood.

•       The solution addresses some, but not all of the mathematical components presented in the task.

•       The student uses a strategy that is partially useful, and demonstrates some evidence of mathematical reasoning.

•       Some parts of the project may be correct, but major errors are noted and the student could not completely carry out mathematical procedures.

•       Your project is hard to follow because the material is presented in a manner that jumps around between unconnected topics.

•       There is some use of appropriate mathematical representation.

•       There is some use of mathematical terminology and notation appropriate for the problem.

•       Your project contains low quality graphics and colors that do not add interest to the project.

1

•       There is no solution, or the solution has no relationship to the task.

•       No evidence of a strategy, procedure, or mathematical reasoning and/or uses a strategy that does not help solve the problem.

•       The solution addresses none of the mathematical components presented in the task.

•       There were so many errors in mathematical procedures that the problem could not be solved.

•       There is no explanation of the solution, the explanation cannot be understood or it is unrelated to the problem.

•       There is no use, or inappropriate use, of mathematical representations (e.g. figures, diagrams, graphs, tables, etc.).

•       There is no use, or mostly inappropriate use, of mathematical terminology and notation.

•       Your project is missing graphics and uses little to no color.

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