Results 1 to 3 of 3

Math Help - word problem please help!

  1. #1
    Newbie
    Joined
    Nov 2008
    Posts
    1

    word problem please help!

    can anyone please explain to me how you do this word problem?

    baking a tray of corn muffins takes 4 c milk and 3 c wheat flour. A tray of bran muffins takes 2 c of milk and 3 c of flour. A baker has 16 c milk and 15 c wheat flower. He makes 3$ profit per tray of corn muffins and 2$ profit of bran muffins. How many trays of each type of muffin should the baker make to maximize his profit?
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Super Member

    Joined
    May 2006
    From
    Lexington, MA (USA)
    Posts
    11,914
    Thanks
    778
    Hello, freakyfrog!

    Are you skimming through some book?
    This is a Linear Programming problem.
    If this is an assignment, you should have been taught the procedure.


    A tray of corn muffins takes 4 c milk and 3 c wheat flour.
    A tray of bran muffins takes 2 c of milk and 3 c of flour.
    A baker has 16 c milk and 15 c wheat flower.
    He makes $3 profit per tray of corn muffins and $2 profit of bran muffins.
    How many trays of each type of muffin should the baker make to maximize his profit?

    Let x = number of trays of corn muffins, x \geq 0
    Let y = number of trays of bran muffins, y \geq 0


    Place the information in a chart:

    . . \begin{array}{c||c|c||c}<br />
& \text{milk} & \text{flour} & \text{Profit} \\ \hline<br />
\text{corn }(x) & 4x & 3x & 3x \\<br />
\text{bran }(y) & 2y & 3y & 2y \\ \hline<br />
\text{Total} & 16 & 15 \end{array}


    We have: . \begin{array}{cccccccc}4x + 2y & \leq & 16 & \Rightarrow & 2x + y & \leq & 8 & [1] \\<br />
3x + 3y & \leq & 15 & \Rightarrow & x + y &\leq & 5 & [2] \end{array}

    [1] Graph the line: 2x + y \:=\:8
    . . Intercepts: (4,0), (0,8)
    . . Shade the region below the line.

    [2] Graph the line: x + y \:=\:5
    . . Intercepts: (5,0), (0,5)
    . . Shade the region below the line.

    The shaded region looks like this:
    Code:
          |
        8 *
          |*
          | *
        5 o  *
          |:* *
          |:::**
          |:::::o
          |::::::**
          |::::::::* *
          |::::::::*  *
      - - o - - - - o - * - -
                    4   5

    We are concerned with only the vertices of the shaded region.

    We see three of them: (0,0), (4,0), (0,5)

    The fourth is the intersection of the two lines.
    Solve the system and we get: (3,2)


    Test these vertices in the profit function: . P \:=\:3x+2y
    . . and see which one produces maximum profit.


    . . \begin{array}{c|c} & \text{Profit} \\ \text{Vertex} & 3x+2y \\ \hline<br />
(0,0) & 0 \\ (4,0) & 12 \\ (0,5) & 10 \\ {\color{blue}(3,2)} & {\color{blue}13} \end{array}


    For maximum profit, he should make 3 trays of corn muffins and 2 trays of bran muffins.

    Follow Math Help Forum on Facebook and Google+

  3. #3
    A riddle wrapped in an enigma
    masters's Avatar
    Joined
    Jan 2008
    From
    Big Stone Gap, Virginia
    Posts
    2,551
    Thanks
    12
    Awards
    1
    Quote Originally Posted by freakyfrog View Post
    can anyone please explain to me how you do this word problem?

    baking a tray of corn muffins takes 4 c milk and 3 c wheat flour. A tray of bran muffins takes 2 c of milk and 3 c of flour. A baker has 16 c milk and 15 c wheat flower. He makes 3$ profit per tray of corn muffins and 2$ profit of bran muffins. How many trays of each type of muffin should the baker make to maximize his profit?
    This is a linear programming model. We need to set up a set of constraints, graph them and locate the vertices of the feasible region to find the right combination to maximize profit.

    Let x = no. of corn muffins to make
    Let y = no. of bran muffins to make

    (1) x\ge 0
    (2) y\ge 0
    (3) 4x + 2y \leq 16\Longrightarrow y\leq8-2x
    (4) 3x+3y \leq 15\Longrightarrow y\leq5-x

    Profit Function= P(x,y)=3x+2y

    I graphed the inequalities on a TI-84+ and isolated the feasible region. The vertices are (4, 0), (0, 0), (0, 5), (3, 2)

    Substitute these points into your profit function and see which set makes P(x, y) the largest.

    I don't know what tools you are allowed to use. If you had to do this without a calculator, you would have to find the intersections of the lines using your favorite method of solving simultaneous linear equations.

    Good luck.

    EDIT: Soroban, even with all that code you write, I'm too slow.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. word problem - the snow plough problem?
    Posted in the Differential Equations Forum
    Replies: 1
    Last Post: October 13th 2011, 02:02 PM
  2. Replies: 3
    Last Post: January 2nd 2011, 09:20 PM
  3. 1 Word problem and 1 function problem
    Posted in the Algebra Forum
    Replies: 8
    Last Post: April 21st 2010, 09:01 AM
  4. Replies: 2
    Last Post: January 10th 2009, 06:49 AM

Search Tags


/mathhelpforum @mathhelpforum