Question

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?

Answer 1

What you need to do in this problem is find an equation for the
temperature \[\theta \left( t \right)\]
First rewrite it so you can integrate it like so

dθ/(Ta−θ(t))=10dt

now integrate it
\[\ln | \theta(t)-Ta | = -10t + C\]
You will notice I made each side the negative, so it's smoother to isolate the temperature function.
Now exponentiate each side
\[\theta \left( t \right) - Ta = e ^{-10t+C} \] now set some letter, say 'A' equal to \[e ^{c}\] and because
\[e^{-10t+C} \] = \[e^{-10t}e^{C}\]
this \[\theta \left( t \right) - Ta = e ^{-10t+C} \]
becomes
\[\theta \left( t \right) - Ta = Ae ^{-10t} \]
From here it is elementary to find all the answers to the questions you were asked, you were given the initial temperature and the temperature constant of the room so you can deduce the value of 'A'. If this doesn't make sense go back and review differentials, also review the natural logarithm, base e and their derivatives.