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2:51 padlet earsznov 3mo Newton's Law of Cooling "A mathematician is a device for turning coffee into theorems." - Alfréd Rényi www Background A container of hot water at temperature, T, placed in a room of lower temperature Troom, will result in an exchange of heat from the hot water to the room. The water will eventually cool to the same temperature as the room. You observe this cooling process every time you wait for a hot drink to cool. In this experiment, you will examine the cooling of hot water, with the goal of creating a model that describes the process. You can also predict the time it takes for the hot water to cool to room temperature. Isaac Newton modeled the cooling process by assuming that the rate at which thermal energy moved from one body to another is proportional (by a constant k that depends padlet.com AA
2:51 7 the goal of creating a model that describes the process. You can also predict the time it takes for the hot water to cool to room temperature. Isaac Newton modeled the cooling process by assuming that the rate at which thermal energy moved from one body to another is proportional (by a constant k that depends on the object and the mode of heat transfer) to the difference in temperature between the two bodies, Tdiff. In the case of a sample of water cooling at room temperature air cooling rate = -kTdiff From this simple assumption, he showed that the temperature change is exponential in time and can be predicted by T = Toe T+ Troom where To is the initial temperature difference. Exponential changes are common in science. When a rate of change AA padlet.com
2:51 7 that the temperature change is exponential in time and can be predicted by T = Toe-Taft + Troom where To is the initial temperature difference. Exponential changes are common in science. When a rate of change is proportional to the changing quantity, the behavior is exponential. Preliminary Questions What are some other real-world applications of exponential functions? Use the internet to find two or more additional examples. A Dilemma A coffee drinker is faced with the following dilemma. She is not going to drink her hot coffee with cream for ten minutes but wants it to still be as hot as possible. Is it better to immediately add the room- temperature cream, stir the coffee, and let it sit for ten minutes, or is it better to let the coffee sit for ten minutes and then add and AA padlet.com
2:51 A Dilemma A coffee drinker is faced with the following dilemma. She is not going to drink her hot coffee with cream for ten minutes but wants it to still be as hot as possible. Is it better to immediately add the room- temperature cream, stir the coffee, and let it sit for ten minutes, or is it better to let the coffee sit for ten minutes and then add and stir in the cream? Which results in a higher temperature after ten minutes? Adding Cream Immediately Time 0-1 min 1-2 min 2-3 min 72.3 3-4 min 63.8 4-5 min 57.0 5-6 min 51.4 7-8 min 46.8 8-9 min 43.0 9-10 min 39.9 Temperature (begin) "C Temperature (end) "C Rate of Cooling "Cin 87.6 86.9 82.1 80.1 70.5 62.3 TRoom = 23.7 °C AA 55.8 50.4 46.0 42.3 39.3 Waiting Ten Minutes to Add padlet.com
2:52 5-6 min 51.4 7-8 min 46.8 8-9 min 43.0 9-10 min 39.9 TRoom = 23.7 °C Waiting Ten Minutes to Add Cream Time 0-1 min 1-2 min 2-3 min 3-4 min 4-5 min 5-6 min 7-8 min 8-9 min 37.1 9-10 min 34.2 TRoom Temperature (begin) "C Temperature (end) "C Rate of Cooling "Cimin 90.7 86.6 80.2 77.3 67.3 65.1 57.6 56.0 50.4 49.2 AA 50.4 46.0 42.3 39.3 44.9 40.5 = 25.0 °C 43.9 39.8 36.5 33.7 Activity Make a prediction with justification to address the coffee drinker's dilemma outlined above. Record your prediction. Use Desmos to plot the data. Apply a best- fit of the form: 3₁~ Ae-B²1+ C padlet.com
2:52 Activity Make a prediction with justification to address the coffee drinker's dilemma outlined above. Record your prediction. Use Desmos to plot the data. Apply a best- fit of the form: Y₁ Ae B+C Develop an evidence-based conclusion and explain your findings in terms of Newton's law of cooling. Assignment Create a GRAPHIC, FOCUSED, and ORDERED science poster illustrating your approach and principal findings. Your science poster should include the following: (1) Title (2) Research question(s). (3) Methodology (clearly describe your approach). (4) Data, Results, and Analysis. (5) An evidence-based conclusion statement. Post your work to Seesaw. AA padlet.com
2:52 Develop an evidence-based conclusion and explain your findings in terms of Newton's law of cooling. Y₁~ Aе-B²1 +C Assignment Create a GRAPHIC, FOCUSED, and ORDERED science poster illustrating your approach and principal findings. Your science poster should include the following: (1) Title (2) Research question(s). (3) Methodology (clearly describe your approach). (4) Data, Results, and Analysis. (5) An evidence-based conclusion statement. Post your work to Seesaw. Solution The addition of the cream early reduces the difference in temperature between the coffee and the air, so it reduces the rate of heat loss during the waiting period. Following Newton's law of cooling, the cup stays hotter if the cream is added first. < AA padlet.com