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Related rates weight of duckling

  1. May 5, 2013 #1
    1. The problem statement, all variables and given/known data
    Suppose that during the first year after its hatching, the weight of a duck increases at a rate proportional to its weight. The duck weighed 2 pounds when hatched, and 3.5 lbs at age 4 months. How many lbs will it weight at 6 months?

    A) 4.2 lbs
    B) 4.6 lbs
    C) 4.8 lbs
    D) 5.6 lbs
    E) 6.5 lbs

    Answer: (B) 4.6 lbs

    2. Relevant equations



    3. The attempt at a solution

    I assumed this was a linear equation and used the points (0, 2) and (4, 3.5) to find the slope, .375, and the equation of the line to be y=.375x+2. Then i plugged in 6 and got y(6)=.375(6)+2 = 4.25. But the answer is 4.6.

    Is the graph not a linear equation? Maybe since it says "weight increases at a rate proportional to its weight", it is an exponential function?? Then, how would I find f(6)??
     
  2. jcsd
  3. May 5, 2013 #2

    rock.freak667

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    If dw/dt is the rate of change of weight, then

    dw/dt ∝ w such that dw/dt = kw where k is a constant.

    You will need to solve this DE to get w(t).

    Then they gave you two conditions, so that you can solve for your constants.
     
  4. May 5, 2013 #3
    What do you mean, dw/dt=kw? by using (0,2) and (4,3.5), how would i solve for w(t)??
     
  5. May 5, 2013 #4

    SteamKing

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    Read the problem statement carefully. "the weight of a duck increases at a rate proportional to its weight."

    Thus, let w(t) = the weight of the duck at time t.

    the rate of change in weight with respect to time is dw/dt.

    This rate is proportional to the weight of the duck, thus dw/dt = kw, where k = constant of proportionality.

    Do you know about separation of variables?
     
  6. May 5, 2013 #5
    ok, I see. So i took dw/dt = kw and got
    ∫1/w dw = ∫k dt
    lnw = kt + C

    Using (0,2) to find C:
    ln2 = C

    Using (4, 3.5) to find k:
    ln3.5 = 4.5k + ln2
    4.5k = ln(1.75)
    k=1.124

    So, lnw= 0.124t + ln2
    w=2e^(0.124t)


    to plug in 6 months:
    w(6)= 4.218

    How come I didnt get the correct answer, 4.6 lbs?
     
  7. May 5, 2013 #6

    haruspex

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    2016 Award

    4 became 4.5?
     
  8. May 5, 2013 #7
    Thank you
     
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