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Calculus Word Problem - Thermodynamics

  1. Jun 26, 2012 #1
    Consider an object at temperature θ in an a place which temperature is Ta. The rate of change of the temperature is given as: dθ/dt = 10(Ta − θ(t)). If the room temperature is constant at Ta=20,and the initial temperature of the object is θ(0) = 100. When will the object reach temperatures of 60,40 and 30?

    Attempt:

    Integrating both sides of equation dθ/dt = 10(Ta − θ(t)) , but i am struggling with that. I am not sure if it is θ = 10 * Ta * t - 5 θˆ2
     
    Last edited: Jun 26, 2012
  2. jcsd
  3. Jun 26, 2012 #2
    The given equation is,

    [tex]\frac{d\theta}{dt} = 10(Ta − θ(t))[/tex]

    Where θ(t) presumably denotes the temperature at a given time t. Dividing both the equations by [itex](Ta − θ(t)[/itex] you will have,

    [tex]\frac{d\theta}{(Ta − θ)} = 10dt[/tex]

    This you can integrate to find the relation between time and temperature.
     
  4. Jun 26, 2012 #3
    Thanks! Integrating this do i get -log(Ta-θ) = 10 * t ?
     
  5. Jun 26, 2012 #4
    Yes! :smile:

    You can also think about integrating,

    [tex]\frac{d\theta}{\theta - Ta} = -10dt[/tex]

    Because it turns out easier to evaluate.
     
  6. Jun 27, 2012 #5
    You left out the constant of integration. You need to use the initial condition θ =100 at t = 0.
     
  7. Jun 27, 2012 #6
    Thanks! Yes I realized i was missing something, but I finally got it right. Thanks everyone!
     
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