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Math Help - Functions & Applications - Help

  1. #1
    Rob
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    Functions & Applications - Help

    Hi, I have been trying to work on this problem and can't seem to figure it out. Please help me with this problem. Thanks

    Problem:


    The cost to produce 50 hard-shell camera cases is $1500,
    and the cost to produce
    80 of the cases is $1800. Each of the cases
    sells for $
    40. If all the cases produced can be sold, and that both cost
    and revenue are given by linear functions, find the cost function, the
    revenue function, graph them and show the break-even point, and
    (with work shown) find the break-even point.


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  2. #2
    is up to his old tricks again! Jhevon's Avatar
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    Quote Originally Posted by Rob View Post
    Hi, I have been trying to work on this problem and can't seem to figure it out. Please help me with this problem. Thanks

    Problem:


    The cost to produce 50 hard-shell camera cases is $1500,

    and the cost to produce
    80 of the cases is $1800. Each of the cases

    sells for $40. If all the cases produced can be sold, and that both cost
    and revenue are given by linear functions, find the cost function, the
    revenue function, graph them and show the break-even point, and
    (with work shown) find the break-even point.




    We are told that these are linear functions, so we know they will be of the form: y = mx + b, where m is the slope and b is the y-intercept.

    Let C(x) be the cost function and R(x) be the revenue function, where x is the number of units sold.

    Revenue is the money recieved from selling x units, that is, the revenue is 40x, since we get 40 dollars for each unit we sell. So:

    R(x) = 40x

    Finding C(x) is a bit more challenging. First we need the slope. we are told the cost to produce 50 hard-shell camera cases is $1500, and the cost to produce 80 of the cases is $1800.

    That is C(50) = 1500 and C(80) = 1800
    => (x1,y1) = (50, 1500) and (x2,y2) = (80, 1800)
    using the slope formula:
    m = (y2 - y1)/(x2 - x1) = (1800 - 1500)/(80 - 50) = 300/30 = 10

    now we use either of the points and our calculated value for m and plug them into the point-slope form:
    Using (x1,y1) = (50,1500), m = 10
    y - y1 = m(x - x1)
    => y - 1500 = (10)(x - 50)
    => y = 10x - 500 + 1500
    => y = 10x + 1000
    => C(x) = 10x + 1000


    The break even point is where the revenue equals the cost of production, that is, where:

    10x + 1000 = 40x
    => 30x = 1000
    => x = 33 1/3
    so after 33 1/3 units are sold, we break even

    Attached Thumbnails Attached Thumbnails Functions & Applications - Help-cost.jpg  
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  3. #3
    Rob
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    Thank you! this helps me alot.
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  4. #4
    is up to his old tricks again! Jhevon's Avatar
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    Quote Originally Posted by Rob View Post
    Thank you! this helps me alot.
    You're welcome
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