Project #8239 - Exponential Model

Exponential Regression (ER)

Data: A cup of hot coffee was placed in a room maintained at a constant temperature of 69 degrees.
 

Thermometer
 
Measuring Coffee Temperature
 

The temperature of the coffee was recorded periodically, and the following table was compiled.
 

Table 1:
 

Time Elapsed (minutes)    Coffee Temperature (deg F)
x T
0 166
10 140.5
20 125.2
30 110.3
40 104.5
50 98.4
60 93.9

REMARKS: Common sense tells us that the coffee will be cooling off and its temperature will decrease and approach the ambient temperature of the room, 69 degrees.

So, the temperature difference between the coffee temperature and the room temperature will decrease to 0.

We will be fitting the data to an exponential curve of the form y = A e- bx. Notice that as x gets large, y will get closer and closer to 0, which is what the temperature difference will do.

So, we want to analyze the data where x = time elapsed and y = T - 69, the temperature difference between the coffee temperature and the room temperature.
 

Table 2:
 

Time Elapsed (minutes)    Temperature Difference (deg F)
x y = T - 69
0 97
10 71.5
20 56.2
30 41.3
40 35.5
50 29.4
60 24.9

 

Tasks for Exponential Regression Model (ER)

(ER-1) Plot the points (x, y) in the second table (Table 2) to obtain a scatterplot. Note that the trend is definitely non-linear. Use an appropriate scale on the horizontal and vertical axes and be sure to label carefully.

(ER-2) Find the exponential function of best fit and graph it on the scatterplot. State the formula for the exponential function. It should have the form y = A e- bx where software has provided you with the numerical values for A and b.

(ER-3) Find and state the value of r2, the coefficient of determination. Discuss your findings.(r2 is calculated using a different formula than for linear regression. However, just as in the linear case, the closer r2 is to 1, the better the fit.) Is an exponential curve a good curve to fit to this data?

(ER-4) Use the exponential function to make a coffee temperature estimate. Each class member will compute a temperature estimate for a different elapsed time. Look at the Class Members list in our Webtycho classroom, and look for the number "n" next to your name. Take your number n, multiply it by 6  to get an elapsed time x; that is, compute x = 6n. The value of x is the time (in minutes) you will substitute into the exponential function to make a temperature estimate.

For instance, if your n is 14, then x = 6(14) = 84 minutes, and then you would substitute x = 84 minutes into your exponential function y = A e- bx to get y, the corresponding temperature difference between the coffee temperature and the room temperature. Since y = T - 69, we have coffee temperature T = y + 69. Take your y estimate and add 69 degrees to get the coffee temperature estimate. State your results clearly -- the elapsed time and the corresponding estimate of the coffee temperature.

(ER-5) Use the exponential function together with algebra to estimate the elapsed time when the coffee arrived at a particular target temperature.  Report the time to the nearest tenth of a minute. Each class member will work with a different target temperature. Take your "n" value from ER-4 and add it to 69 to get your target temperature T.
 

 For instance, if your n is 14, then your target temperature T is 69 + 14 = 83 degrees, and your goal is to use the exponential function find out how long it took the coffee to cool to 83 degrees. Note that 14 is the temperature difference between the coffee and the room, what we are calling y. So, for this particular target temperature, the goal is finding how long it took for the temperature difference y to reach 14 degrees; that is, solving the equation 14 = A e- bx for x

 In general, you are solving your equation y = A e- bx for x, where y = your particular temperature difference. Show algebraic work in solving your equation. State your results clearly -- your target temperature and the estimated elapsed time, to the nearest tenth of a minute.
 

Your completed project must include your name. You may submit all of your project in one document or a combination of documents, which may consist of word processing documents or spreadsheets or scanned handwritten work, provided it is clearly labeled where each checklist item can be found.  Projects are graded on the basis of completeness, correctness, ease in locating all of the checklist items, and overall strength of the presentation.

Technology Tips

To complete the Exponential Model portion of the project, you will need to use technology (or hand-drawing) to create a scatterplot, find a regression curve, plot the regression curve, and find r2.

For the given set of data, you will carry out exponential regression.

Below are some options, together with some videos. Each video is limited to 5 minutes or less. It takes a bit of time for the video to initially download. When playing the video, if you want to slow it down to read the text, hit the pause icon. (If you run the mouse over the bottom of the video screen, the video controls will appear.)  You may need to adjust the volume.

The basic options are to:

    (1) Generate by hand and scan. 

    (2) Use Microsoft Excel.

Visit  Exponential Regression (VIDEO) to see how to create a scatter plot and use exponential regression with Microsoft Excel. (Create the scatterplot in the same way as for the linear regression, and just choose a different model to get exponential regression.)

    (3) Use Open Office.

    (4) Use a hand-held graphing calculator (See section 1.4 in your textbook for help with Texas Instruments hand-held calculators. MyMathLab also has some guides to calculator resources.)

    (5) Use an online tool 

The free online graphing calculator at http://www.meta-calculator.com/online/  (see picture)  allows you to find the exponential curves and create a scatter plot. For more on this tool, see the Technology Tips for the Linear Model portion of the project.

The result of the free tool is not as nice looking as the Microsoft Excel version, but it is free.

Subject Mathematics
Due By (Pacific Time) 06/28/2013 05:00 pm
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