A coffee with temperature $LaTeX: \displaystyle 154^\circ$ is left in a room with temperature $LaTeX: \displaystyle 59^\circ$ . After 7 minutes the temperature of the coffee is $LaTeX: \displaystyle 149^\circ$ , how long until the coffee is $LaTeX: \displaystyle 124^\circ$ ?

Newton's law of Cooling states that the change in temperature is directly proportional to the difference between the object's temperature and its surroundings. $LaTeX: \frac{dT}{dt} = k(T(t)-T_{\text{room}})$ Using the substitution $LaTeX: \displaystyle y(t)=T(t)-59$ and calculating the derivative gives $LaTeX: \displaystyle \frac{dy}{dt}=\frac{dT}{dt}$ . Calculating the new initial condition using the point $LaTeX: \displaystyle (7, 149)$ and the substition gives $LaTeX: \displaystyle y(0) = T(0)-59 = 95$ . The point $LaTeX: \displaystyle (7, 149)$ must also be transformed to get $LaTeX: \displaystyle y(7) = T(7)-59 = 149 - 59 = 90$ . Substituting both of these into the equation gives the new equaiton $LaTeX: \displaystyle \frac{dy}{dt}=ky$ which has the solution $LaTeX: \displaystyle y(t) = y(0)e^{kt}=95e^{kt}$ . Evaluating the function at the point gives $LaTeX: \displaystyle 90=95e^{7k}$ and isolating the exponential gives $LaTeX: \displaystyle \frac{18}{19}=e^{7k}$ . Solving for $LaTeX: \displaystyle k$ gives $LaTeX: \displaystyle k=\frac{\ln{\left(\frac{18}{19} \right)}}{7}$ . Substuting $LaTeX: \displaystyle k$ back into the equation gives $LaTeX: \displaystyle y(t) = 95e^{\frac{\ln{\left(\frac{18}{19} \right)}}{7}t}$ and simplifying gives $LaTeX: \displaystyle y(t) = 95 \left(\frac{18}{19}\right)^{\frac{t}{7}}$ . Substituting out $LaTeX: \displaystyle y(t)$ gives $LaTeX: T(t)-59 = 95 \left(\frac{18}{19}\right)^{\frac{t}{7}} \implies\, T(t)= 95 \left(\frac{18}{19}\right)^{\frac{t}{7}} + 59$ Using $LaTeX: \displaystyle T$ gives the equation $LaTeX: \displaystyle 124=95 \left(\frac{18}{19}\right)^{\frac{t}{7}} + 59$ . Isolating the exponential gives $LaTeX: \displaystyle \frac{13}{19}=\left(\frac{18}{19}\right)^{\frac{t}{7}}$ . Taking the natural logarithm of both sides and solving for $LaTeX: \displaystyle t$ gives $LaTeX: \displaystyle t = \frac{7 \ln{\left(\frac{13}{19} \right)}}{\ln{\left(\frac{18}{19} \right)}}\approx 49.1$ minutes.