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

Download \(\LaTeX\)

\begin{question}A coffee with temperature $170^\circ$ is left in a room with temperature $85^\circ$. After 7 minutes the temperature of the coffee is $145^\circ$, what is the temperature of the coffee after 14 minutes?
    \soln{9cm}{Using $T = T_0+(T_1-T_0)e^{kt}$ gives $T = 85+(170-85)e^{kt}= 85+85e^{kt}$. Using the point $(7, 145)$ gives $145= 85+85e^{k(7)}$. Isolating the exponential gives $\frac{12}{17}=e^{7k}$. Solving for $k$ gives $k=\frac{\ln{\left(\frac{12}{17} \right)}}{7}$.  Substuting $k$ back into the equation gives $T = 85+85e^{\frac{\ln{\left(\frac{12}{17} \right)}}{7}t}$ and simplifying gives $T = 85 \left(\frac{12}{17}\right)^{\frac{t}{7}} + 85$. Using $t = 14$ gives $T =85 \left(\frac{12}{17}\right)^{\frac{14}{7}} + 85\approx 127^\circ$}

\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 170^\circ " src="/equation_images/%20%5Cdisplaystyle%20170%5E%5Ccirc%20" alt="LaTeX:  \displaystyle 170^\circ " data-equation-content=" \displaystyle 170^\circ " />  is left in a room with temperature  <img class="equation_image" title=" \displaystyle 85^\circ " src="/equation_images/%20%5Cdisplaystyle%2085%5E%5Ccirc%20" alt="LaTeX:  \displaystyle 85^\circ " data-equation-content=" \displaystyle 85^\circ " /> . After 7 minutes the temperature of the coffee is  <img class="equation_image" title=" \displaystyle 145^\circ " src="/equation_images/%20%5Cdisplaystyle%20145%5E%5Ccirc%20" alt="LaTeX:  \displaystyle 145^\circ " data-equation-content=" \displaystyle 145^\circ " /> , what is the temperature of the coffee after 14 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 = 85+(170-85)e^{kt}= 85+85e^{kt} " src="/equation_images/%20%5Cdisplaystyle%20T%20%3D%2085%2B%28170-85%29e%5E%7Bkt%7D%3D%2085%2B85e%5E%7Bkt%7D%20" alt="LaTeX:  \displaystyle T = 85+(170-85)e^{kt}= 85+85e^{kt} " data-equation-content=" \displaystyle T = 85+(170-85)e^{kt}= 85+85e^{kt} " /> . Using the point  <img class="equation_image" title=" \displaystyle (7, 145) " src="/equation_images/%20%5Cdisplaystyle%20%287%2C%20145%29%20" alt="LaTeX:  \displaystyle (7, 145) " data-equation-content=" \displaystyle (7, 145) " />  gives  <img class="equation_image" title=" \displaystyle 145= 85+85e^{k(7)} " src="/equation_images/%20%5Cdisplaystyle%20145%3D%2085%2B85e%5E%7Bk%287%29%7D%20" alt="LaTeX:  \displaystyle 145= 85+85e^{k(7)} " data-equation-content=" \displaystyle 145= 85+85e^{k(7)} " /> . Isolating the exponential gives  <img class="equation_image" title=" \displaystyle \frac{12}{17}=e^{7k} " src="/equation_images/%20%5Cdisplaystyle%20%5Cfrac%7B12%7D%7B17%7D%3De%5E%7B7k%7D%20" alt="LaTeX:  \displaystyle \frac{12}{17}=e^{7k} " data-equation-content=" \displaystyle \frac{12}{17}=e^{7k} " /> . 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{12}{17} \right)}}{7} " src="/equation_images/%20%5Cdisplaystyle%20k%3D%5Cfrac%7B%5Cln%7B%5Cleft%28%5Cfrac%7B12%7D%7B17%7D%20%5Cright%29%7D%7D%7B7%7D%20" alt="LaTeX:  \displaystyle k=\frac{\ln{\left(\frac{12}{17} \right)}}{7} " data-equation-content=" \displaystyle k=\frac{\ln{\left(\frac{12}{17} \right)}}{7} " /> .  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 = 85+85e^{\frac{\ln{\left(\frac{12}{17} \right)}}{7}t} " src="/equation_images/%20%5Cdisplaystyle%20T%20%3D%2085%2B85e%5E%7B%5Cfrac%7B%5Cln%7B%5Cleft%28%5Cfrac%7B12%7D%7B17%7D%20%5Cright%29%7D%7D%7B7%7Dt%7D%20" alt="LaTeX:  \displaystyle T = 85+85e^{\frac{\ln{\left(\frac{12}{17} \right)}}{7}t} " data-equation-content=" \displaystyle T = 85+85e^{\frac{\ln{\left(\frac{12}{17} \right)}}{7}t} " />  and simplifying gives  <img class="equation_image" title=" \displaystyle T = 85 \left(\frac{12}{17}\right)^{\frac{t}{7}} + 85 " src="/equation_images/%20%5Cdisplaystyle%20T%20%3D%2085%20%5Cleft%28%5Cfrac%7B12%7D%7B17%7D%5Cright%29%5E%7B%5Cfrac%7Bt%7D%7B7%7D%7D%20%2B%2085%20" alt="LaTeX:  \displaystyle T = 85 \left(\frac{12}{17}\right)^{\frac{t}{7}} + 85 " data-equation-content=" \displaystyle T = 85 \left(\frac{12}{17}\right)^{\frac{t}{7}} + 85 " /> . Using  <img class="equation_image" title=" \displaystyle t = 14 " src="/equation_images/%20%5Cdisplaystyle%20t%20%3D%2014%20" alt="LaTeX:  \displaystyle t = 14 " data-equation-content=" \displaystyle t = 14 " />  gives  <img class="equation_image" title=" \displaystyle T =85 \left(\frac{12}{17}\right)^{\frac{14}{7}} + 85\approx 127^\circ " src="/equation_images/%20%5Cdisplaystyle%20T%20%3D85%20%5Cleft%28%5Cfrac%7B12%7D%7B17%7D%5Cright%29%5E%7B%5Cfrac%7B14%7D%7B7%7D%7D%20%2B%2085%5Capprox%20127%5E%5Ccirc%20" alt="LaTeX:  \displaystyle T =85 \left(\frac{12}{17}\right)^{\frac{14}{7}} + 85\approx 127^\circ " data-equation-content=" \displaystyle T =85 \left(\frac{12}{17}\right)^{\frac{14}{7}} + 85\approx 127^\circ " /> </p> </p>