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Using \(\displaystyle T = T_0+(T_1-T_0)e^{kt}\) gives \(\displaystyle T = 72+(169-72)e^{kt}= 72+97e^{kt}\). Using the point \(\displaystyle (6, 156)\) gives \(\displaystyle 156= 72+97e^{k(6)}\). Isolating the exponential gives \(\displaystyle \frac{84}{97}=e^{6k}\). Solving for \(\displaystyle k\) gives \(\displaystyle k=\frac{\ln{\left(\frac{84}{97} \right)}}{6}\). Substuting \(\displaystyle k\) back into the equation gives \(\displaystyle T = 72+97e^{\frac{\ln{\left(\frac{84}{97} \right)}}{6}t}\) and simplifying gives \(\displaystyle T = 97 \left(\frac{84}{97}\right)^{\frac{t}{6}} + 72\). Using \(\displaystyle t = 11\) gives \(\displaystyle T =97 \left(\frac{84}{97}\right)^{\frac{11}{6}} + 72\approx 147^\circ\)

Download \(\LaTeX\)

\begin{question}A coffee with temperature $169^\circ$ is left in a room with temperature $72^\circ$. After 6 minutes the temperature of the coffee is $156^\circ$, what is the temperature of the coffee after 11 minutes?
    \soln{9cm}{Using $T = T_0+(T_1-T_0)e^{kt}$ gives $T = 72+(169-72)e^{kt}= 72+97e^{kt}$. Using the point $(6, 156)$ gives $156= 72+97e^{k(6)}$. Isolating the exponential gives $\frac{84}{97}=e^{6k}$. Solving for $k$ gives $k=\frac{\ln{\left(\frac{84}{97} \right)}}{6}$.  Substuting $k$ back into the equation gives $T = 72+97e^{\frac{\ln{\left(\frac{84}{97} \right)}}{6}t}$ and simplifying gives $T = 97 \left(\frac{84}{97}\right)^{\frac{t}{6}} + 72$. Using $t = 11$ gives $T =97 \left(\frac{84}{97}\right)^{\frac{11}{6}} + 72\approx 147^\circ$}

\end{question}

Download Question and Solution Environment\(\LaTeX\)

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\begin{document}\begin{question}(10pts) The question goes here!
    \soln{9cm}{The solution goes here.}

\end{question}\end{document}

HTML for Canvas

<p> <p>A coffee with temperature  <img class="equation_image" title=" \displaystyle 169^\circ " src="/equation_images/%20%5Cdisplaystyle%20169%5E%5Ccirc%20" alt="LaTeX:  \displaystyle 169^\circ " data-equation-content=" \displaystyle 169^\circ " />  is left in a room with temperature  <img class="equation_image" title=" \displaystyle 72^\circ " src="/equation_images/%20%5Cdisplaystyle%2072%5E%5Ccirc%20" alt="LaTeX:  \displaystyle 72^\circ " data-equation-content=" \displaystyle 72^\circ " /> . After 6 minutes the temperature of the coffee is  <img class="equation_image" title=" \displaystyle 156^\circ " src="/equation_images/%20%5Cdisplaystyle%20156%5E%5Ccirc%20" alt="LaTeX:  \displaystyle 156^\circ " data-equation-content=" \displaystyle 156^\circ " /> , what is the temperature of the coffee after 11 minutes?</p> </p>

HTML for Canvas

<p> <p>Using  <img class="equation_image" title=" \displaystyle T = T_0+(T_1-T_0)e^{kt} " src="/equation_images/%20%5Cdisplaystyle%20T%20%3D%20T_0%2B%28T_1-T_0%29e%5E%7Bkt%7D%20" alt="LaTeX:  \displaystyle T = T_0+(T_1-T_0)e^{kt} " data-equation-content=" \displaystyle T = T_0+(T_1-T_0)e^{kt} " />  gives  <img class="equation_image" title=" \displaystyle T = 72+(169-72)e^{kt}= 72+97e^{kt} " src="/equation_images/%20%5Cdisplaystyle%20T%20%3D%2072%2B%28169-72%29e%5E%7Bkt%7D%3D%2072%2B97e%5E%7Bkt%7D%20" alt="LaTeX:  \displaystyle T = 72+(169-72)e^{kt}= 72+97e^{kt} " data-equation-content=" \displaystyle T = 72+(169-72)e^{kt}= 72+97e^{kt} " /> . Using the point  <img class="equation_image" title=" \displaystyle (6, 156) " src="/equation_images/%20%5Cdisplaystyle%20%286%2C%20156%29%20" alt="LaTeX:  \displaystyle (6, 156) " data-equation-content=" \displaystyle (6, 156) " />  gives  <img class="equation_image" title=" \displaystyle 156= 72+97e^{k(6)} " src="/equation_images/%20%5Cdisplaystyle%20156%3D%2072%2B97e%5E%7Bk%286%29%7D%20" alt="LaTeX:  \displaystyle 156= 72+97e^{k(6)} " data-equation-content=" \displaystyle 156= 72+97e^{k(6)} " /> . Isolating the exponential gives  <img class="equation_image" title=" \displaystyle \frac{84}{97}=e^{6k} " src="/equation_images/%20%5Cdisplaystyle%20%5Cfrac%7B84%7D%7B97%7D%3De%5E%7B6k%7D%20" alt="LaTeX:  \displaystyle \frac{84}{97}=e^{6k} " data-equation-content=" \displaystyle \frac{84}{97}=e^{6k} " /> . Solving for  <img class="equation_image" title=" \displaystyle k " src="/equation_images/%20%5Cdisplaystyle%20k%20" alt="LaTeX:  \displaystyle k " data-equation-content=" \displaystyle k " />  gives  <img class="equation_image" title=" \displaystyle k=\frac{\ln{\left(\frac{84}{97} \right)}}{6} " src="/equation_images/%20%5Cdisplaystyle%20k%3D%5Cfrac%7B%5Cln%7B%5Cleft%28%5Cfrac%7B84%7D%7B97%7D%20%5Cright%29%7D%7D%7B6%7D%20" alt="LaTeX:  \displaystyle k=\frac{\ln{\left(\frac{84}{97} \right)}}{6} " data-equation-content=" \displaystyle k=\frac{\ln{\left(\frac{84}{97} \right)}}{6} " /> .  Substuting  <img class="equation_image" title=" \displaystyle k " src="/equation_images/%20%5Cdisplaystyle%20k%20" alt="LaTeX:  \displaystyle k " data-equation-content=" \displaystyle k " />  back into the equation gives  <img class="equation_image" title=" \displaystyle T = 72+97e^{\frac{\ln{\left(\frac{84}{97} \right)}}{6}t} " src="/equation_images/%20%5Cdisplaystyle%20T%20%3D%2072%2B97e%5E%7B%5Cfrac%7B%5Cln%7B%5Cleft%28%5Cfrac%7B84%7D%7B97%7D%20%5Cright%29%7D%7D%7B6%7Dt%7D%20" alt="LaTeX:  \displaystyle T = 72+97e^{\frac{\ln{\left(\frac{84}{97} \right)}}{6}t} " data-equation-content=" \displaystyle T = 72+97e^{\frac{\ln{\left(\frac{84}{97} \right)}}{6}t} " />  and simplifying gives  <img class="equation_image" title=" \displaystyle T = 97 \left(\frac{84}{97}\right)^{\frac{t}{6}} + 72 " src="/equation_images/%20%5Cdisplaystyle%20T%20%3D%2097%20%5Cleft%28%5Cfrac%7B84%7D%7B97%7D%5Cright%29%5E%7B%5Cfrac%7Bt%7D%7B6%7D%7D%20%2B%2072%20" alt="LaTeX:  \displaystyle T = 97 \left(\frac{84}{97}\right)^{\frac{t}{6}} + 72 " data-equation-content=" \displaystyle T = 97 \left(\frac{84}{97}\right)^{\frac{t}{6}} + 72 " /> . Using  <img class="equation_image" title=" \displaystyle t = 11 " src="/equation_images/%20%5Cdisplaystyle%20t%20%3D%2011%20" alt="LaTeX:  \displaystyle t = 11 " data-equation-content=" \displaystyle t = 11 " />  gives  <img class="equation_image" title=" \displaystyle T =97 \left(\frac{84}{97}\right)^{\frac{11}{6}} + 72\approx 147^\circ " src="/equation_images/%20%5Cdisplaystyle%20T%20%3D97%20%5Cleft%28%5Cfrac%7B84%7D%7B97%7D%5Cright%29%5E%7B%5Cfrac%7B11%7D%7B6%7D%7D%20%2B%2072%5Capprox%20147%5E%5Ccirc%20" alt="LaTeX:  \displaystyle T =97 \left(\frac{84}{97}\right)^{\frac{11}{6}} + 72\approx 147^\circ " data-equation-content=" \displaystyle T =97 \left(\frac{84}{97}\right)^{\frac{11}{6}} + 72\approx 147^\circ " /> </p> </p>