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Math Help - i need Quick help with inequalities word problems! only 2!

  1. #1
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    i need Quick help with inequalities word problems! only 2!

    (1 pt) A company manufactures two models of snowboards, standard and deluxe. Each deluxe model requires 26 hours to produce, and 96 units of material. Each standard model requires 13 hours to produce and 80 units of material. The company has 949 production hours available and 4368 units of material in stock. The deluxe model sells for $238 and the standard model sells for $170.

    What amount of each model should be produced and sold to obtain the maximum revenue?
    What is the maximum revenue? $
    ______________________________

    An accounting firm has 1350 hours of staff time and 322 hours of reviewing time available each week. The firm charges $140 for an audit and $35 for a tax return. Each audit requires 90 hours of staff time and 14 hours of review time. Each tax return requires 15 hours of staff time and 7 hours of review time.

    What number of audits and tax returns will yield the maximum revenue?
    What is the maximum revenue?
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  3. #3
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    Ok, this kinda seems a bit like simplex to me, so thats what i did.


    first one:

    let x equal the amount needed to make a standard snowboard
    let y equal the amount needed to make a deluxe snowboard

    Maximise P = 170x + 238y
    subject to
    13x + 26y </= 949 (hours needed)
    80x + 96y </= 4368 (units of material needed)

    put it through the simplex algorithm and you should get
    P = $10,064, when x = 27, and y = 23



    second one:

    let x equal the amount of audits received
    let y equal the amount of tax returns received

    Maximise P = 140x + 35y
    subject to
    90x + 15y </= 1350 (hours of staff time)
    14x + 7y </= 322 (hours of review time)


    Again, put it through simplex and you should get

    P = 2380, when x = 11, and y = 24



    I think this is right at least
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  4. #4
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    Hello, offthewake!

    These are "linear programming" problems.
    I'll baby-step through the second one . . .


    An accounting firm has 1350 hours of staff time
    and 322 hours of reviewing time available each week.
    The firm charges $140 for an audit and $35 for a tax return.
    Each audit requires 90 hours of staff time and 14 hours of review time.
    Each tax return requires 15 hours of staff time and 7 hours of review time.

    What number of audits and tax returns will yield the maximum revenue?
    What is the maximum revenue?

    Make a table of the data . . .

    . . \begin{array}{c||c|c|}<br />
& \text{Staff} & \text{Review} \\ \hline \hline<br />
\text{Audits} & 90 & 14 \\ \hline<br />
\text{Taxes} & 15 & 7 \\ \hline  \end{array}



    Let: . \begin{array}{ccccc}x &=& \text{no. of audits,} & x \geq 0 & [1]\\ y &=& \text{no. of taxes,} & y \geq 0 & [2] \end{array}

    The table becomes:

    . . \begin{array}{c||c|c|}<br />
& \text{Staff} & \text{Review} \\ \hline \hline<br />
\text{Audits} & 90x & 14x \\ \hline<br />
\text{Taxes} & 15y & 7y \\ \hline \hline<br />
\text{Total} & 1350 & 322 \end{array}

    . . Revnue: . R \;=\;140x + 35y


    Reading down the columns, we find our inequalities:

    . . \begin{array}{cccccccc}90x + 15y & \leq 1350 & \Rightarrow & 6x + y & \leq & 90 & [3] <br />
\\ 14x + 7y & \leq 322 & \Rightarrow & 2x + y & \leq & 46 & [4] \end{array}


    Now we graph the four inequalities.


    [1] and [2] places us in Quadrant 1.


    To graph 6x + y \:\leq\:90, graph the line: . 6x + y \:{\color{red}=}\:90
    . . and shade the region below it.
    Its intercepts are: (15,0) and (0,90).
    . . Plot the intercepts, draw the line, shade the region below the line.

    To graph 2x + y \:\leq \:46, graph the line: . 2x + y \:{\color{red}=}\:46
    . . and shade the region below it.
    Its intercepts are: (23,0) and (0,46).
    . . Plot the intercepts, draw the line, shade the region below the line.


    The graph will look like this:
    Code:
            |
         90 *
            |*
            | *
            |  *
            |   *
            |    *
         46 o     *
            |:::*  *
            |:::::::o
            |::::::::*  *
            |:::::::::*     *
            |::::::::::*        *
        - - o - - - - - o - - - - - * - -
            0          15          23

    We are concerned with the vertices of the shaded region.
    We can see three of them: . (0,0),\:(15,0),\:(0,46)

    The fourth vertex is the intersection of the two slanted lines.

    Solve the system: . \begin{array}{ccc}6x + y &=& 90 \\ 2x+ y &=& 46\end{array}\quad \Rightarrow\quad \begin{array}{c}x \:=\:11 \\ y \:=\:24 \end{array}

    . . Hence, the fourth vertex is: . (11,24)


    Finally, test the four vertices: . (0,0),\;(14,0),\;(0,46),\;(11,24)
    . . in the Revenue function to see which produces maximum revenue.

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