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Math Help - Rational Function

  1. #1
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    Rational Function

    This problem set involves a formula (a rational function) with which a new tortilla company might be able to forecast its production over the first few weeks of operation.
    In this formula, C(t) is the number of bags of tortillas that can be produced per week after t weeks of production.
    Here is the rational function that was developed to best describe the fledgling company's production.
    C(t) = 1000t^2 - 10,000t
    t^2 - 10t + 25
    1.What does the graph of the function look like? To answer this question, find:
    a) The horizontal asymptote, if any:
    b) The t intercepts, if any:
    c) The C intercept, if any:


    2.What is the projected maximum number of bags of tortillas that the company can never exceed? Discuss this answer in terms of the horizontal asymptote.


    3.Can the company reach that maximum? If so, after how long?

    4.Give an interpretation of what might be happening to the company's production efforts from week 5 to week 10? Discuss this answer in detail.

    5.In terms of the business application, is there any meaning for the value of C(t) when t = -5 and t = -10? Explain your answer.

    This is a big part of my test and I have no clue. Any help at all would be great. Thanks. Also, that is all my questions I do beleive so thanks again.

    Kasey
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  2. #2
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    Quote Originally Posted by flippin4u View Post
    This problem set involves a formula (a rational function) with which a new tortilla company might be able to forecast its production over the first few weeks of operation.
    In this formula, C(t) is the number of bags of tortillas that can be produced per week after t weeks of production.
    Here is the rational function that was developed to best describe the fledgling company's production.
    C(t) = 1000t^2 - 10,000t
    t^2 - 10t + 25
    1.What does the graph of the function look like? To answer this question, find:
    a) The horizontal asymptote, if any:
    b) The t intercepts, if any:
    c) The C intercept, if any:
    ...
    Hello,

    to a) You get the horizontal asymptote by calculating the limit:
    <br />
\lim_{t \to \infty}\left(\frac{1000t^2-10000t}{t^2-10t+25} \right)= \lim_{t\to\infty}\left(\frac{t^2\left(1000-\frac{10000}{t}\right)}{t^2\left(1-\frac{10}{t}+\frac{25}{t^2}\right)} \right)=1000

    to b) You'll get the t-intercepts if C(t) = 0. That means the numerator of the fraction must be zero:

    1000t^2-10000t = 0~\iff~t = 0~\vee~t = 10

    to c) You'll get the C-intercepts if t = 0. Plug in t = 0 into the given equation and you have C(0) = 0

    Quote Originally Posted by flippin4u View Post
    2.What is the projected maximum number of bags of tortillas that the company can never exceed? Discuss this answer in terms of the horizontal asymptote.
    Calcultae the first derivative of C(t):

    C'(t) = \frac{5000}{(t-5)^3} As you can see this function is positive for t > 5 that means the graph of C is increasing. The graph of C approaches the horizontal asymptote from below but never reaches the value of 1000.

    I've attached a diagram with the vertical (red) and horizontal (blue) asymptotes.
    Attached Thumbnails Attached Thumbnails Rational Function-tortilla_production.gif  
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  3. #3
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    Quote Originally Posted by flippin4u View Post
    This problem set involves a formula (a rational function) with which a new tortilla company might be able to forecast its production over the first few weeks of operation.
    In this formula, C(t) is the number of bags of tortillas that can be produced per week after t weeks of production.
    Here is the rational function that was developed to best describe the fledgling company's production.
    C(t) = 1000t^2 - 10,000t
    t^2 - 10t + 25
    ...
    2.What is the projected maximum number of bags of tortillas that the company can never exceed? Discuss this answer in terms of the horizontal asymptote.


    3.Can the company reach that maximum? If so, after how long?

    ...
    Hello,

    one additional remark:

    As I've shown the maximum amount of bags is 999. To answer the question when the company will reach this amount you have to solve the following equation for t:

    C(t) = 999~\implies~\frac{1000t^2-10000t}{t^2-10t+25}=999 multiply by denominator ==>  1000t^2-10000t = 999 \cdot (t^2-10t+25)

    Expand the bracket and collect all terms at the LHS of the equation:

    t^2-10t-24975 = 0 solve for t.

    I've got: t = 163.11...~\vee~t = -153.11...

    That means: From week #164 on the company produces allways 999 bags per week. (The productions never reaches 1000 bags/week)
    Maybe these calculations answer the questions about the maximum production.
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