Page 1 of 2 12 LastLast
Results 1 to 15 of 24
Like Tree2Thanks

Math Help - exponential growth

  1. #1
    Newbie
    Joined
    Nov 2012
    From
    Australia
    Posts
    15

    exponential growth

    Can anyone explain more about exponential growth of the concentration on nutrients in an estuary when water is flowing into and out of the estuary at the same rate?
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor MarkFL's Avatar
    Joined
    Dec 2011
    From
    St. Augustine, FL.
    Posts
    1,988
    Thanks
    734

    Re: exponential growth

    Are you given initial conditions and the concentration of nutrients flowing in?
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Newbie
    Joined
    Nov 2012
    From
    Australia
    Posts
    15

    Re: exponential growth

    the flow in rate is 100m^3/day with concentration of 90g/m^3.
    Initial concentration in the estuary is 20g/m^3
    vol 1000m^3
    flow out rate is also 100m^3/day
    Follow Math Help Forum on Facebook and Google+

  4. #4
    MHF Contributor MarkFL's Avatar
    Joined
    Dec 2011
    From
    St. Augustine, FL.
    Posts
    1,988
    Thanks
    734

    Re: exponential growth

    First, you want to determine the mass of nutrient initially present. Amount per volume times volume equals amount. What do you find?
    Follow Math Help Forum on Facebook and Google+

  5. #5
    Newbie
    Joined
    Nov 2012
    From
    Australia
    Posts
    15

    Re: exponential growth

    20g/m^3 x 1000m^3 = 20000g/m^3 initial concentration
    and I have 90 x 100/day = 9000 flowing in
    Follow Math Help Forum on Facebook and Google+

  6. #6
    MHF Contributor MarkFL's Avatar
    Joined
    Dec 2011
    From
    St. Augustine, FL.
    Posts
    1,988
    Thanks
    734

    Re: exponential growth

    You have the correct initial mass of 20,000 g, but you don't want to call this a concentration, it is simply an amount. Also, you are correct that 9000 grams is flowing in. So, let's then use smaller numbers and say the initial mass is 20 kg and 9 kg is flowing in per day.

    Let's let N(t) represent the mass of nutrient present in the estuary at time t.

    Now, we know the time rate of change of N(t) is equal to the rate in minus the rate out. We have already determined the rate in. The rate out will be a function of N(t). We will asume the concentration of nutrient is uniform in the estuary. That is, the concentration of nutrient in any part of the estuary at time t is just N(t) divided by the volume of fluid in the estuary. Since the flow in is equal to the flow out, this volume remains constant at 1000\text{ m}^3.

    Hence, the output rate of nutrient is:

    \left(100\frac{\text{m}^3}{\text{day}} \right)\left(\frac{N(t)}{1000}\,\frac{\text{kg}}{ \text{m}^3} \right)=\frac{N(t)}{10}\,\frac{\text{kg}}{ \text{day}}

    So, we now have enough information to model N(t) with the IVP:

    \frac{dN}{dt}=9-\frac{N(t)}{10} where N(0)=20

    Can you proceed from here?
    Follow Math Help Forum on Facebook and Google+

  7. #7
    Newbie
    Joined
    Nov 2012
    From
    Australia
    Posts
    15

    Re: exponential growth

    would this mean that on day zero, my flow out amount is 9 -20/10 = 7?
    we are supposed to write down and solve an appropriate differential equation for N(t) along with the appropriate initial condition. That would somehow be my initial amount plus the flow in rate, minus the flow out rate? My apologies for sounding so thick!
    Follow Math Help Forum on Facebook and Google+

  8. #8
    MHF Contributor MarkFL's Avatar
    Joined
    Dec 2011
    From
    St. Augustine, FL.
    Posts
    1,988
    Thanks
    734

    Re: exponential growth

    The initial output rate of nutrient would be 7 kg/day, but that is really only applicable at the very start of the first day, the initial moment.

    I have already stated the appropriate differential equation/initial condition. This is called an initial value problem (IVP).

    Now, to solve the ordinary differential equation (ODE), do you recognize what type of equation it is, and if so, what you need to do to solve it?
    Follow Math Help Forum on Facebook and Google+

  9. #9
    MHF Contributor MarkFL's Avatar
    Joined
    Dec 2011
    From
    St. Augustine, FL.
    Posts
    1,988
    Thanks
    734

    Re: exponential growth

    By the way, are you supposed to find the mass of nutrient at time t, or the concentration? Since the concentration is just mass/volume and the volume is constant it is simple to convert between the two, I was just curious.
    Follow Math Help Forum on Facebook and Google+

  10. #10
    Newbie
    Joined
    Nov 2012
    From
    Australia
    Posts
    15

    Re: exponential growth

    we are supposed to find the amount at time t and then the concentration after a long time
    Follow Math Help Forum on Facebook and Google+

  11. #11
    MHF Contributor MarkFL's Avatar
    Joined
    Dec 2011
    From
    St. Augustine, FL.
    Posts
    1,988
    Thanks
    734

    Re: exponential growth

    Okay, good, then we are set to do just that.
    Follow Math Help Forum on Facebook and Google+

  12. #12
    Newbie
    Joined
    Nov 2012
    From
    Australia
    Posts
    15

    Re: exponential growth

    it's an exponential growth equation.
    I need the initial amount plus the flow in amount = 20 +9^t minus the flow out amount
    Follow Math Help Forum on Facebook and Google+

  13. #13
    MHF Contributor MarkFL's Avatar
    Joined
    Dec 2011
    From
    St. Augustine, FL.
    Posts
    1,988
    Thanks
    734

    Re: exponential growth

    No, the ODE is linear, as we may write it in the form:

    \frac{dN}{dt}+\frac{1}{10}N(t)=9

    Do you recognize this form? Do you recall how to transform the equation so that we have the product of a differentiation on the left?
    Follow Math Help Forum on Facebook and Google+

  14. #14
    Newbie
    Joined
    Nov 2012
    From
    Australia
    Posts
    15

    Re: exponential growth

    I'm just reading this from my text book. The equation is already in the required form(dN/dt + P(t)N= Q(t)
    Follow Math Help Forum on Facebook and Google+

  15. #15
    MHF Contributor MarkFL's Avatar
    Joined
    Dec 2011
    From
    St. Augustine, FL.
    Posts
    1,988
    Thanks
    734

    Re: exponential growth

    Yes, so what do we need to do next?
    Follow Math Help Forum on Facebook and Google+

Page 1 of 2 12 LastLast

Similar Math Help Forum Discussions

  1. Exponential growth
    Posted in the Algebra Forum
    Replies: 1
    Last Post: March 22nd 2010, 07:38 AM
  2. Exponential growth
    Posted in the Algebra Forum
    Replies: 3
    Last Post: February 24th 2010, 07:16 PM
  3. exponential growth
    Posted in the Pre-Calculus Forum
    Replies: 1
    Last Post: November 3rd 2008, 04:17 PM
  4. exponential growth
    Posted in the Calculus Forum
    Replies: 1
    Last Post: December 14th 2006, 12:00 AM
  5. Exponential Growth.
    Posted in the Calculus Forum
    Replies: 1
    Last Post: November 23rd 2006, 11:35 AM

Search Tags


/mathhelpforum @mathhelpforum