A coffee with temperature $LaTeX: \displaystyle 170^\circ$ is left in a room with temperature $LaTeX: \displaystyle 85^\circ$ . After 7 minutes the temperature of the coffee is $LaTeX: \displaystyle 145^\circ$ , what is the temperature of the coffee after 14 minutes?

Using $LaTeX: \displaystyle T = T_0+(T_1-T_0)e^{kt}$ gives $LaTeX: \displaystyle T = 85+(170-85)e^{kt}= 85+85e^{kt}$ . Using the point $LaTeX: \displaystyle (7, 145)$ gives $LaTeX: \displaystyle 145= 85+85e^{k(7)}$ . Isolating the exponential gives $LaTeX: \displaystyle \frac{12}{17}=e^{7k}$ . Solving for $LaTeX: \displaystyle k$ gives $LaTeX: \displaystyle k=\frac{\ln{\left(\frac{12}{17} \right)}}{7}$ . Substuting $LaTeX: \displaystyle k$ back into the equation gives $LaTeX: \displaystyle T = 85+85e^{\frac{\ln{\left(\frac{12}{17} \right)}}{7}t}$ and simplifying gives $LaTeX: \displaystyle T = 85 \left(\frac{12}{17}\right)^{\frac{t}{7}} + 85$ . Using $LaTeX: \displaystyle t = 14$ gives $LaTeX: \displaystyle T =85 \left(\frac{12}{17}\right)^{\frac{14}{7}} + 85\approx 127^\circ$