Results 1 to 2 of 2

Thread: [SOLVED] Exponential models

  1. #1
    friend37866
    Guest

    [SOLVED] Exponential models

    The temperature in a room is 21 degrees Celsius. A thermometer which has been kept in it is placed outside. After 5 minutes the thermometer reading is 16 Celsius. Five minutes later, it is 13 Celsius. Find the outside temperature.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Forum Admin topsquark's Avatar
    Joined
    Jan 2006
    From
    Wellsville, NY
    Posts
    11,152
    Thanks
    731
    Awards
    1
    Quote Originally Posted by friend37866 View Post
    The temperature in a room is 21 degrees Celsius. A thermometer which has been kept in it is placed outside. After 5 minutes the thermometer reading is 16 Celsius. Five minutes later, it is 13 Celsius. Find the outside temperature.
    This is related to Newton's Law of Cooling, I presume. (See part way down the page of this link.)

    We have that
    $\displaystyle T(t) = T_{env} + (T(0) - T_{env})e^{-t/t_0}$
    where $\displaystyle T(0)$ is the temperature of the body (in this case the thermometer), $\displaystyle T_{env}$ is the temperature of the environment, and $\displaystyle t_0$ is the "time constant," which we don't know yet.

    We know that $\displaystyle T(0) = 21~^oC$ and that at $\displaystyle t = 5~min$, $\displaystyle T(5) = 16~^oC$, and at $\displaystyle t = 10~min$, $\displaystyle T(10) = 13~^oC$.

    So:
    $\displaystyle T(5) = 16 = T_{env} + (21 - T_{env})e^{-5/t_0}$
    and
    $\displaystyle T(10) = 13 = T_{env} + (21 - T_{env})e^{-10/t_0}$

    Two equations and two unknowns. We want $\displaystyle T_{env}$ and don't care about the time constant, so what I recommend you do is, instead of trying to solve for $\displaystyle t_0$ directly, solve the top equation for $\displaystyle e^{-5/t_0}$. Then note that $\displaystyle e^{-10/t_0} = \left ( e^{-5/t_0} \right )^2$ and use that for your substitution. You'll be left with an equation for $\displaystyle T_{env}$.

    Sorry, I gotta run.

    -Dan
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Exponential models
    Posted in the Pre-Calculus Forum
    Replies: 2
    Last Post: Mar 10th 2010, 01:28 AM
  2. Exponential Models and Application help
    Posted in the Algebra Forum
    Replies: 3
    Last Post: Nov 9th 2009, 06:29 AM
  3. Exponential Models
    Posted in the Algebra Forum
    Replies: 3
    Last Post: Mar 25th 2009, 04:23 PM
  4. More Exponential Models
    Posted in the Algebra Forum
    Replies: 1
    Last Post: Mar 25th 2009, 02:28 PM
  5. Logarithms and exponential models
    Posted in the Pre-Calculus Forum
    Replies: 2
    Last Post: Mar 9th 2009, 05:53 PM

Search Tags


/mathhelpforum @mathhelpforum