# Linear Programming – Great Real-world Applications

I’ve been playing around with graphing inequalities on the Casio Prizm (and fx-9860GII & fx-9750GII), realizing how easy they are to graph and how easy it is to find the area of intersection. In doing so, trying to find some relevant problems that would be applications of graphing inequalities, I was reminded how relevant linear programming problems are to both graphing inequalities and finding intersections, but more importantly, to real-world situations.

I spent the majority of my teaching in K-12 with Algebra and Geometry, so linear programming was never on the required curriculum back when I was teaching.  It’s usually in an Algebra II course other ‘higher level’ mathematics courses.  And, if I remember, it is often a content that is ‘skipped’ because it isn’t on standardized tests. In looking at some of the activities in Fostering Algebraic Thinking with Casio Prizm (Goerdt, Horton), trying to find some inequality problems, I found some activities under Linear Programming that made me stop and think – these problems are much more relevant and connect to actual situations than those “naked math” inequality problems we tend to give our students. Find the intersection of these two inequalities is not very engaging, but find out number of servings of potato chips and peanuts you can have while staying within your recommended daily allowance (I know I am asking this question!!). Or, if a company wants to advertise in the SuperBowl, based on their budget and cost of an ad during the first half or the second half, when should they advertise? Still looking for intersections of inequalities but now making a connection in context where there is a purpose for using the math.

These type of activities require a lot more math than just graphing inequalities and finding intersections.  You have to translate the situation into inequalities. You have to convert these inequalities to a form that  will allow you to graph them (i.e. y=). You have to consider ‘implied’ constraints (if you are dealing with money, you can’t have x or y less than zero, for example). There’s a lot of sense-making about what is being asked and the constraints given, modeling a situation with the correct mathematical equation, seeing if your answer makes sense, understanding what the graphical representation of the intersection means, and what do the intersection points mean in the context of the situation.  Linear Programming problems are really robust applications. There’s a lot of Common Core content and more importantly, Mathematical Practices going on here.  Students can then apply this type of thinking to situations that impact them – for example, selling girl scout cookies….if we have to make a certain amount, the cookies cost this amount, the troop gets this percentage, how many do we need to sell to make our profit?  I think these are a lot more interesting than ‘find the intersections of these inequalities’.

I have attached an activity  from Fostering Algebraic Thinking called “The Snack Problem” (Download: Linear Program Snack Prob).  You can look at it in depth by downloading it.  I am just going to state the problem here, identify the inequalities, and then do a quick video on how to graph and find the intersection.  There is a lot more to the activity then whats below so I encourage you to take a look and try it with your students.

Problem:  Assume you like snacks and insist on having at least one serving of dry roasted peanuts and one serving of potato chips each day.  Each serving of the peanuts contains 15% of the recommended daily allowance of saturated fat; each serving of potato chips contains 10%.  Each serving of the peanuts contains 12% of the recommended amount of dietary fiber; each serving of potato chips contains 5%. You determine you want to consume no more than 60% of the recommended allowance of dietary fat from these two snacks, but you want to get at least 30% of the recommended allowance of fiber from them.  Sketch the feasible region.

(please refer to the handout for the other questions).

Inequalities: x represents dry roasted peanuts, y represents potato chips

• 0.15x + 0.10y ≤ 0.60 (saturated fats) –> y≤(0.60-0.15x)/0.10
• 0.21x + 0.05y ≥ 0.30 (fiber)  –>                        y ≥ (0.30-0.21x)/0.05
• x ≥ 1 (at least 1 serving peanuts a day)
• y ≥ 1 (at least 1 serving potato chips a day)

The video below shows how to graph all four of these inequalities on one graph (yes, even x ≥ 1)! And how to construct just the intersection (vs. the union of these four inequalities) and find specific intersection points.  The attached activity includes all these steps as well.