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Math Help - Need a bit of help understanding optimization problem

  1. #1
    Senior Member Paze's Avatar
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    Need a bit of help understanding optimization problem

    Hi MHF.

    I have an optimization problem:

    A farmer can purchase two types of food products for his horses, one is called Lingo and the other is called Bingo.
    Each bag of Lingo contains: 20gr of fats, 30gr of carbs and 5gr of proteins.
    Each bag of Bingo contains: 10gr of fats, 30gr of carbs and 10gr of proteins.

    The farmer needs a minimum of: 460gr of fats, 960 grams of carbs and 220 grams of proteins for his horses.

    Each bag of Lingo costs 3000 and each bag of Bingo costs 3500.

    How many of each bags should the farmer purchase to satisfy the minimum requirements?

    I'm guessing that the constraint is the price, 3000 and 3500? I can do optimization problems with fences (the classic) but this one seems a lot different, but perhaps it isn't?

    A little nudge would be appreciated!
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  2. #2
    MHF Contributor

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    Re: Need a bit of help understanding optimization problem

    No, the prices are NOT the constraints. The constraints are "The farmer needs a minimum of: 460gr of fats, 960 grams of carbs and 220 grams of proteins for his horses."

    You are told that
    "Each bag of Lingo contains: 20gr of fats, 30gr of carbs and 5gr of proteins.
    Each bag of Bingo contains: 10gr of fats, 30gr of carbs and 10gr of proteins."

    So if you buy X bags of Lingo and Y bags of Bingo you are supplying 20X+ 10Y grams of fats, 30X+ 30Y grams of carbs, and 5X+ 10Y grams of proteins. Since, again, "The farmer needs a minimum of: 460gr of fats, 960 grams of carbs and 220 grams of proteins for his horses."
    he must have 20X+ 10Y\ge 460, 30X+ 30Y\ge 960, and 5X+ 10Y\ge 220.

    The question "How many of each bags should the farmer purchase to satisfy the minimum requirements?" doesn't ask anything about cost of the bags.
    I would solve this by graphing 20X+ 10Y= 460, 30X+ 30Y= 960, and 5X+ 10Y= 220, calculating where they intersect and finding the lowest vertex of the region above all the lines.
    Last edited by HallsofIvy; July 20th 2013 at 07:32 AM.
    Thanks from Paze
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  3. #3
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    Re: Need a bit of help understanding optimization problem

    Hello, Paze!

    A farmer can purchase two types of food products for his horses,
    one is called Lingo and the other is called Bingo.

    Each bag of Lingo contains: 20gr of fats, 30gr of carbs and 5gr of proteins.
    Each bag of Bingo contains: 10gr of fats, 30gr of carbs and 10gr of proteins.

    The farmer needs a minimum of:
    . . 460gr of fats, 960 grams of carbs and 220 grams of proteins for his horses.

    Each bag of Lingo costs 3000 and each bag of Bingo costs 3500.

    How many of each bags should the farmer purchase to satisfy the minimum requirements?
    . . and minimize his cost

    Let x = bags of Lingo.
    Let y = bags of Bingo.


    Tabulate the given data:

    . . \begin{array}{c||c|c|c|} & \text{fats} & \text{carbs} & \text{prot.} \\ \hline \hline \text{Lingo }(x) & 20 & 30 & 5 \\ \hline \text{Bingo }(y) & 10 & 30 & 10 \\ \hline \hline \text{Total} & 460 & 960 & 220 \\ \hline \end{array}


    Form the inequalities:

    . . \begin{array}{cccccc}20x + 10y \:\ge\:460 &\Rightarrow&  2x + y \:\ge\:46 \\ \\ 30x + 30y \:\ge\:960 & \Rightarrow &  x+y \:\ge\:32  \\ \\ 5x + 10y \:\ge\:220 &\Rightarrow&  x + 2y \:\ge\:44 \end{array}


    Graph the lines and shade region above them.


    The graph should look like this:

    Code:
          |(0,46)
          o
          |*
          | *
          |  * (14,18) 
          |   o
          |     *
          |       * (20,12)
          |         o
          |             *
          |                 *
          |                     *
      - - + - - - - - - - - - - - - o - -
          |                       (44,0)

    Test the coordinates in the cost function: . C \:=\:3000x + 3500y
    . . and determine which point gives the minimum cost.
    Thanks from Paze
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  4. #4
    MHF Contributor

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    Re: Need a bit of help understanding optimization problem

    Uh, Soroban, the question, as Paze stated it, was "How many of each bags should the farmer purchase to satisfy the minimum requirements?" There is nothing said about "minimum cost"- though it is quite possible that the question should[ have been "How many of each bags should the farmer purchase to satisfy the minimum requirements, at the minimum cost?"
    Thanks from Paze
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  5. #5
    Senior Member Paze's Avatar
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    Re: Need a bit of help understanding optimization problem

    Thanks guys. I have graphed the lines and I come up with the same sort of coordinates as Soroban. I understand the functions that I am using but I'm a bit stuck where you tell me to "Test the coordinates in the cost function". Also, is graphing the only way to go? I'm pretty sure I won't be allowed to use graphing calculators during the test.
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  6. #6
    Senior Member Paze's Avatar
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    Re: Need a bit of help understanding optimization problem

    Quote Originally Posted by HallsofIvy View Post
    Uh, Soroban, the question, as Paze stated it, was "How many of each bags should the farmer purchase to satisfy the minimum requirements?" There is nothing said about "minimum cost"- though it is quite possible that the question should[ have been "How many of each bags should the farmer purchase to satisfy the minimum requirements, at the minimum cost?"
    Yes, that is what the question should have stated. Sorry...I'm translating these questions (poorly).
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  7. #7
    Senior Member Paze's Avatar
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    Re: Need a bit of help understanding optimization problem

    Perhaps I am still not understanding this wholly. What exactly am I looking for in this picture? Which part gives me the information that I am looking for?Need a bit of help understanding optimization problem-whattoread.png
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  8. #8
    Senior Member Paze's Avatar
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    Re: Need a bit of help understanding optimization problem

    I think I understand the problem now.

    If I wanted to do this 100% algebraically I would find the 3 intersecting points and then test them in the cost function as so:

    c=3000x+3500y

    c1=3000*14+3500*18=105000

    c2=3000*20+3500*12=102000

    c3=3000*16+3500*14=97000

    Buying 16 Lingo and 14 Bingo is the cheapest option that fulfills the requirements, yes?

    It's a little different than my usual optimization problem where I need to differentiate to find high/low values though so that's what caught me off guard here.

    Am I correct in my solution? Thank you.
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