Calculus Word Problem - Thermodynamics

brunocamba

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

Infinitum

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.

brunocamba

Thanks! Integrating this do i get -log(Ta-θ) = 10 * t ?

Infinitum

Yes!

You can also think about integrating,

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

Because it turns out easier to evaluate.

Chestermiller

You left out the constant of integration. You need to use the initial condition θ =100 at t = 0.

brunocamba

Thanks! Yes I realized i was missing something, but I finally got it right. Thanks everyone!