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Math Help - Linear programming world problem; system of inequalities

  1. #1
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    Linear programming world problem; system of inequalities

    A hospital dietician wishes. To prepare a corn-squash vegetable dish that will provide at least 3 grams of protein and cost no more that .36 per serving. And ounce of cream corn provides 1/2 gram of protein and costs .04. An ounce of squash supplies 1/4 gram of protein and costs .03. For taste, there must be at least 2 ounces of corn and at least as much squash as corn. It is important to keep the total number of ounces in a serving as small as possible. Find the combination of corrn and squash that will minimize the amount of ingrediants used per serving. I am horrible at word problems and I've been trying to get the hang of it, please help
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  2. #2
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    Re: Linear programming world problem; system of inequalities

    Here is what you do. First you must define your variables. What do you want to be X and what do you want to be Y? I chose X= cream corn and Y=squash. (Keep in mind you do not need to use X and Y, I just always use them). Next you see you will have a constraint for grams of protein, cost per serving, and taste. Keep in mind that X=> 0 and Y=> 0 almost always, as long as there is no negative. You start reading and you see proteins. Cream Corn provides 1/2 gram of protein and Squash provides 1/4 gram of protein and the dish needs to provide atleast 3 grams of protein (Hint: atleast, must be =>) so the 3rd constraint is 1/2X+1/4Y=>3. Now you see cost per serving. An ounce of Cream Corn costs .04 and an ounce of Squash costs .03 and the dish can not cost over .36. So that means .04X+.03Y<=.36 . Keep on reading... it says ,"For taste, there must be at least 2 ounces of corn and at least as much squash as corn" So x=>2 for the ounces, and for the amount of squash and corn it will be Y=>X because there must be equal or more squash than corn. Now that there are all our constraints/inequalities, you must solve for Y for every equation so that it may fit in your graphing calculator... basic algebra so i don't think i need to show you [Keep in mind that you must have atleast the TI-83plus version so that you can install the Inequalz app on it so that you may have X1 to X6 and the ability to change the equals to inequality signs... to do that, go to apps and turn on inequalz, then go to Y=, then go on thr equal sign, press alpha then press the button,(Y=, WINDOW, ZOOM, TRACE, or GRAPH) that is under the symbol you want. then it will automatically change the sign for you.] Once graphed, you will need the corner points of the feasible region, and the feasible region is the shaded area in the middle of all the lines. (Note: not all the line intersections are the corner points, so to adjust your sight of the feasible region, you might need to zoom in, out, or zbox it sometimes) Once graphed, it will be a mess... so to fix the mess, you will press alpha then Y= then press 1 (this comes with the inequalz app) then it will regraph showing only the place where all shades intersect, AKA your feasible region. Next (another inequalz thing) to find your corner points, you will see POI which means point of intersection... which will show you all the intersections of lines (again, not all line intersections are corner points so watch out) so t activate POI, press Alpha then zoom. After that, you will see all the lines' intersections... its hard to choose a specfic intersection so you will have to press the arrows a while in order to recoord your data points. Once all the corner points are reached, plug them into you objective function ( and for this case i cant find it because this problem is weird... or i am just bad). If your question asks for the max, the largest number will be your answer, if it asks for the min, the smallest number will be your answer. For this, your final answer will be written in a sentence so this is how it would go... "The minimum cost of ingredients is (insert smallest number here) when you buy (X data point) number of Cream Corn ounces and (Y data point) number of Squash ounces" (keep in mind that the X= ounces of cream corn and Y= ounces of squash so make sure you match the data point with the defined variable) Agaain on a test or quiz, you will need to do show your object function which is USUALLY and equation, not an inequality. so if it was maximize profit if its $6 per sold ounce of cream corn and $4 per sold ounce of squash (which is ridiculous but whatever) then your objective function would be P(which is profit)= 6X+4Y. Then plug in your corner points and find the largest number then write your sentence. I hope this helps... P.S- i have a 91% average in my Accelerated Geometry Class and last semester i finished with a 90% average.
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