Question: A historian of science studies the cooling of a 17th-century thermometer, modeled by $ T(t) = 20 + 80e^{-0.1t} $. Find the time $ t $ when the temperature reaches 40°C.

["Understanding 17th-Century Thermometry: Solving for Cooling Time Using $ T(t) = 20 + 80e^{-0.1t} $", "In the study of early scientific instrumentation, one fascinating example involves the cooling behavior of 17th-century thermometers. Historians of science analyze historical temperature models to understand how early scientists interpreted thermal changes. A classic model for such cooling is given by the equation:", "[\nT(t) = 20 + 80e^{-0.1t}\n]", "where $ T(t) $ is the temperature (in °C) at time $ t $ (in minutes), $ 20^\circ $C is the ambient temperature, and $ 80 $ and $ 0.1 $ reflect the initial temperature and cooling rate observed during early experimental studies.", "### How Does This Model Reflect Historical Experimentation?", "During the 1600s, pioneers like Galileo and Fahrenheit laid groundwork in thermometry by noticing that temperatures decreased predictably over time. Though thermometers of the period lacked digital precision, early measurements hinted at exponential decay—precely captured here by the function $ T(t) = 20 + 80e^{-0.1t} $. Solving for when $ T(t) = 40^\circ $°C allows historians and students to quantitatively reconstruct the thermal reasoning of early scientists.", "### Solving the Equation: When Does $ T(t) = 40^\circ $?", "We solve for $ t $ when:", "[\n40 = 20 + 80e^{-0.1t}\n]", "Subtract 20 from both sides:", "[\n20 = 80e^{-0.1t}\n]", "Divide both sides by 80:", "[\n\frac{1}{4} = e^{-0.1t}\n]", "Take the natural logarithm of both sides:", "[\n\ln\left(\frac{1}{4}\right) = -0.1t\n]", "Recall that $ \ln\left(\frac{1}{4}\right) = -\ln 4 $, so:", "[\n-\ln 4 = -0.1t\n]", "Multiply both sides by $-1$:", "[\n\ln 4 = 0.1t\n]", "Solve for $ t $:", "[\nt = \frac{\ln 4}{0.1} = 10 \ln 4\n]", "Since $ \ln 4 = \ln(2^2) = 2 \ln 2 \approx 2 \ imes 0.6931 = 1.3862 $, we compute:", "[\nt \approx 10 \ imes 1.3862 = 13.862\n]", "### Conclusion: The Cooling Moment Near 14 Minutes", "Thus, the temperature reaches approximately $ 40^\circ $C about 13.86 minutes after the start of cooling. This value offers historians a concrete numerical benchmark, illustrating how early scientists might have estimated thermal dynamics using available mathematical models.", "For those studying the history of science, this problem bridges theory and experiment: how a simple exponential function mirrors centuries of observational science. The timeline derived here enriches our understanding of the precision—and limitations—behind 17th-century thermometry.", "---", "Key Takeaways:", "- The cooling model $ T(t) = 20 + 80e^{-0.1t} $ reflects empirical cooling behavior studied during the scientific revolution.\n- Solving $ T(t) = 40 $ reveals $ t = \frac{\ln 4}{0.1} \approx 13.86 $ minutes.\n- This calculation helps historians quantify historical thermal observations and appreciate early scientific reasoning."]