A coffee with temperature $LaTeX: \displaystyle 163^\circ$ is left in a room with temperature $LaTeX: \displaystyle 72^\circ$ . After 4 minutes the temperature of the coffee is $LaTeX: \displaystyle 145^\circ$ , how long until the coffee is $LaTeX: \displaystyle 125^\circ$ ?

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 $LaTeX: \displaystyle y(t)=T(t)-72$ and calculating the derivative gives $LaTeX: \displaystyle \frac{dy}{dt}=\frac{dT}{dt}$ . Calculating the new initial condition using the point $LaTeX: \displaystyle (4, 145)$ and the substition gives $LaTeX: \displaystyle y(0) = T(0)-72 = 91$ . The point $LaTeX: \displaystyle (4, 145)$ must also be transformed to get $LaTeX: \displaystyle y(4) = T(4)-72 = 145 - 72 = 73$ . Substituting both of these into the equation gives the new equaiton $LaTeX: \displaystyle \frac{dy}{dt}=ky$ which has the solution $LaTeX: \displaystyle y(t) = y(0)e^{kt}=91e^{kt}$ . Evaluating the function at the point gives $LaTeX: \displaystyle 73=91e^{4k}$ and isolating the exponential gives $LaTeX: \displaystyle \frac{73}{91}=e^{4k}$ . Solving for $LaTeX: \displaystyle k$ gives $LaTeX: \displaystyle k=\frac{\ln{\left(\frac{73}{91} \right)}}{4}$ . Substuting $LaTeX: \displaystyle k$ back into the equation gives $LaTeX: \displaystyle y(t) = 91e^{\frac{\ln{\left(\frac{73}{91} \right)}}{4}t}$ and simplifying gives $LaTeX: \displaystyle y(t) = 91 \left(\frac{73}{91}\right)^{\frac{t}{4}}$ . Substituting out $LaTeX: \displaystyle y(t)$ gives $LaTeX: T(t)-72 = 91 \left(\frac{73}{91}\right)^{\frac{t}{4}} \implies\, T(t)= 91 \left(\frac{73}{91}\right)^{\frac{t}{4}} + 72$ Using $LaTeX: \displaystyle T$ gives the equation $LaTeX: \displaystyle 125=91 \left(\frac{73}{91}\right)^{\frac{t}{4}} + 72$ . Isolating the exponential gives $LaTeX: \displaystyle \frac{53}{91}=\left(\frac{73}{91}\right)^{\frac{t}{4}}$ . Taking the natural logarithm of both sides and solving for $LaTeX: \displaystyle t$ gives $LaTeX: \displaystyle t = \frac{4 \ln{\left(\frac{53}{91} \right)}}{\ln{\left(\frac{73}{91} \right)}}\approx 9.8$ minutes.