A coffee with temperature $LaTeX: \displaystyle 179^\circ$ is left in a room with temperature $LaTeX: \displaystyle 51^\circ$ . After 3 minutes the temperature of the coffee is $LaTeX: \displaystyle 173^\circ$ , what is the temperature of the coffee after 12 minutes?

Using $LaTeX: \displaystyle T = T_0+(T_1-T_0)e^{kt}$ gives $LaTeX: \displaystyle T = 51+(179-51)e^{kt}= 51+128e^{kt}$ . Using the point $LaTeX: \displaystyle (3, 173)$ gives $LaTeX: \displaystyle 173= 51+128e^{k(3)}$ . Isolating the exponential gives $LaTeX: \displaystyle \frac{61}{64}=e^{3k}$ . Solving for $LaTeX: \displaystyle k$ gives $LaTeX: \displaystyle k=\frac{\ln{\left(\frac{61}{64} \right)}}{3}$ . Substuting $LaTeX: \displaystyle k$ back into the equation gives $LaTeX: \displaystyle T = 51+128e^{\frac{\ln{\left(\frac{61}{64} \right)}}{3}t}$ and simplifying gives $LaTeX: \displaystyle T = 128 \left(\frac{61}{64}\right)^{\frac{t}{3}} + 51$ . Using $LaTeX: \displaystyle t = 12$ gives $LaTeX: \displaystyle T =128 \left(\frac{61}{64}\right)^{\frac{12}{3}} + 51\approx 157^\circ$