As soon as a hot cup of coffee is poured, it begins to cool. The cooling process is rapid at first, then levels off. After a long period of time, the temperature of the coffee eventually reaches room temperature. Temperature variations for such cooling objects were summarized by Newton. He stated that the rate at which a warm body cools is approximately proportional to the temperature difference between the temperature of the warm object and the temperature of its surroundings. Stated mathematically:

where y represents the object's temperature, x represents time, C is the surrounding temperature, and k is a proportionality constant. The following data was collected with the CBL system by heating a piece of aluminum foil and allowing it to cool over a period of three minutes (temperature was measured in degrees Celsius and time in seconds).

Equipment Required:

Equipment Setup Procedure:

  1. Connect the CBL unit to the TI-85 calculator with the unit-to-unit link cable using the I/O ports located on the bottom edge of each unit. Press the cable ends in firmly.
  2. Connect the temperature probe to Channel 1 (CH1) on the top edge of the CBL unit.
  3. Prepare the solution of object to be cooled.
  4. Turn on the CBL unit and the calculator.
    The CBL system is now ready to receive commands from the calculator.

Experiment Procedure:

  1. Fold the 10" side of the aluminum foil in half. Place the temperature probe between the layers of he foil in the middle of one side. Flatten the foil tightly around the probe.
  2. Make sur the CBL is turned on. Start the HEAT program on the TI-85. Enter 3 at the "HOW MUCH TIME BETWEEN POINTS..." prompt.
  3. At the "HOW MUCH TIME FOR TOTAL DATA..." prompt, enter 3. The program records 60 points (one point every 3 seconds for 3 minutes).
  4. Record room temperature in degrees Celsius. Enter the room temperature at the "ROOM TEMP=?" prompt. After you enter room temperature, wait to press ENTER until told to do so.
  5. Turn the hair dryer on high and heat the center of the foil. If the foil blows around in circles, place the nozzle of the dryer closer to the foil. Be sure to keep the hand holding the probe away from the heat.
  6. When the foil is hot (about 70°C), press ENTER on the TI-85 to start collecting data. Turn the hair dryer off and allow the foil to cool. A collected data point is displayed in real-time once every three seconds for three minutes.
  7. Analyze the data.

Analysis and Conclusion:

Part I:

  1. Note that the difference equation given by Newton's Law of Cooling is given by

    Solve this equation for k.
  2. A natural question to ask is whether your data does indeed satisfy this equation. To find out, use the data you collected to find several values for k over one second intervals. What do you notice about these values of k?
  3. The program has also calcuated the value of k over one second intervals. These values are stored in the list DC. You can scroll through this list and compare those values to the values you found in Step 2. The average of all of the values in DC is stored under the name CON. Recall this average by selecting 2ND RCL CON on the home screen of your calculator. How does this average compare to the k you found?

Part II:

  1. Discuss the expected shape of the curve representing the collected data. Will the data points ever reach y=0? What is the lowest temperature that will be recorded?
  2. In this experiment, time data is stored in the list L3, and temperature data is stored in L4. Lower all the recorded temperatures by the room temperature recorded from the classroom thermometer. To do this, go to the home screen and type L4-C STR L5, where C is the room temperature in degrees Celsius. Your new temperature data is now stored in the list L5.
  3. Go to STAT CALC and select lists L3,L5. then choose EXPR to perform an eponential regression. Choose STREG and enter y! at the prompt to store the regression in your first y equation.
  4. Because you subtracted room temperature off of all of your data points, you must now add that back in to your equation. Go to the graphing editor and select the equation y1. Add room temperature to the regression equation.
  5. The resulting equation is of the form y=A*Bx+C. Convert this eqauation to the form y=A*ekx+C. How does the k in this equation compare to the k you found in Part I?
  6. Graph your regression equation along with the Scatter plot of L3,L4.

Part III:

Newton's Law of Cooling is represented by the differential equation

where y(0)=y0>C and y(x1)=y1.
Solve the differential equation to get an equation of the form y=A*ekx+C.

Looking at Part II and Part III, what meaning do you think the variables A and k have in the cooling of an object?

Sample Data

What kind of function should be used to model the above data?

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