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Study Guides > Mathematics for the Liberal Arts

J1.10: Section 4 Part 2

Redefinition of output parameters

            In the example discussed earlier where a soda can warms up in a 75º room, subtraction of the room temperature from the column B temperature-data values was needed in order to use an exponential-decay model to predict the process. This was necessary because the standard exponential-decay model can only converge on zero as its asymptotic value, so when using it we need to model the temperature difference, not the temperature itself. But we can avoid the need for changing the data if we modify the model formula to add on the room temperature to the prediction. The exponential part of the formula will still converge on zero, but the formula as a whole will now converge on the room temperature, as is needed for correct predictions.
Soda-can warm-up in a 75º room
Minutes out of cooler Temp­erature of can
0 35
5 50
10 59
15 65
Example 10: Modify the exponential model formula for the warming of a soda can in a room whose air temperature is 75 ºF, without subtracting 75 from each of the data y values in column B.
  1. Insert a new worksheet and label it “Adjusted soda warm-up model”, then copy into this new worksheet the content of the Exponential Model Template worksheet.
  2. Copy the data into columns A and B, leaving the temperature values as they are.
  3. Set H3 to the phrase “Temperature at 0 minutes” as a reminder of what is being predicted.
  4. Modify the content of cell G1 by adding “+75” at the end just before the last quotation mark.
  5. Modify the content of cell C3 by adding “+75” at the end.
  6. Spread the modified formula in cell C3 down to row 6 to match the data.
  7. Spread the formula in D3 down to row 6.
  8. Make a scatter plot of columns A, B, and C (select the rectangle A1:C6) to show the model and the data together so that you can watch how well they match while you adjust the parameters.
  9. Adjust the model parameters, using the following steps:
    1. Set G3 to 35, which is the dataset value for 0 minutes.
    2. Adjust the G4 growth-rate parameter until the model graph closely follows the shape of the data. The decay rate is the same 8.8% that was found by the subtract-room-temperature method.
    3. If needed for the best fit, make small adjustments to G3 so that the model moves up or down.
  10. Use the model to predict temperature for any specified time (e.g., 20 minutes) by entering the minute number for which you want a prediction at the bottom of column A.
Resulting worksheet:
  A B C D E F G H
1 x y data y model Exponential model: y = a * (1+r)^x + 75
2 Minutes Temperature Prediction y = –40 * (1 – 0.088)^x + 75
3 0 35 35.00 –40 a: Initial temperature difference
4 5 50 49.76 –0.088 r: Growth rate
5 10 59 59.08
6 15 65 64.95

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  • Mathematics for Modeling. Authored by: Mary Parker and Hunter Ellinger. License: CC BY: Attribution.