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Math Help - system of inequalities

  1. #1
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    system of inequalities

    i have this problem

    A stationary company makes two types of notebooks: a deluxe notebook with subject dividers, which sells for $4.00, and a regular notebook, which sells for $3.00. The production cost is $3.20 for each deluxe notebook, and $2.60 for each regular notebook. The company has the facilities to manufacture between 2000 and 3000 deluxe notebooks and between 3000 and 6000 regular notebooks, but not more than 7000 altogether. How many notebooks of each type should be manufactured to maximize the difference between the selling prices and the production costs.

    can someone help me get the system of inequalities i need to use to find the max.
    thanks.
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  2. #2
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    Hello lightninghelix

    Suppose they make x deluxe notebooks and y regular notebooks.
    a deluxe notebook with subject dividers, which sells for $4.00, and a regular notebook, which sells for $3.00.
    The total selling price is $ (4x+3y).
    The production cost is $3.20 for each deluxe notebook, and $2.60 for each regular notebook
    So the total production cost is $ (3.2x+2.6y)
    between 2000 and 3000 deluxe notebooks
    2000 \le x \le 3000
    between 3000 and 6000 regular notebooks
    3000\le y \le 6000
    not more than 7000 altogether
    x+y\le 7000

    Can you continue from here?

    Grandad
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  3. #3
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    so i would graph the lines 4x+3y less than or equal to 7000 and 3.2x+2.6y less than or equal to 7000 ?

    wait i forgot about the objective functions, so i think i would graph all the inequalities you gave then i would use the vertices in those and plug them into both objective functions? which are the production and selling costs?
    Last edited by lightninghelix; April 20th 2009 at 06:47 AM.
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  4. #4
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    Inequalities

    Hello lightninghelix
    Quote Originally Posted by lightninghelix View Post
    so i would graph the lines 4x+3y less than or equal to 7000 and 3.2x+2.6y less than or equal to 7000 ?

    wait i forgot about the objective functions, so i think i would graph all the inequalities you gave then i would use the vertices in those and plug them into both objective functions? which are the production and selling costs?
    Yes. Plot the inequalities and work out the region in which the possible values of x and y lie. (It's a trapezium shape.) Then for the objective function, you need the profit, i.e. difference between the production costs and the selling price; that's

    P = (4x+3y) - (3.2x + 2.6y) = 0.8x + 0.4y

    and find the values of x and y in the region that makes P a maximum. (I think the answer is x = 3000 and y = 4000.)

    Grandad
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  5. #5
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    thanks i dont remember exactly what answer i got right now but i think it was what you put :P
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