1
Algebra II: Strand 6. Exponentials and Logarithms; Topic 2. Applications; Task 6.2.3
TASK 6.2.3: EXPONENTIAL DECAY
Solutions
Part I:
Consider a cup of hot water at approximately 180° F that is placed in a room where the
temperature is 74° F. Think about the cooling effect.
1. How will the water cool over time? Allow time for discussion and various opinions on why a
particular graph is chosen as the model. (see teaching notes)
Graph 3
y
180
150
120
90
60
t
0.5
1.0
1.5
2.0
Time (hours)
2. Describe a situation implied by each graph.
• Will a cup of hot water retain its heat for a while, and then cool rapidly? Which graph
depicts this situation? Graph 3
• Will the water cool rapidly at first and then cool more slowly? Graph 2
• Or, will the cup cool at a constant rate? Graph 1
3, 4, 6 Procedural steps. (see teaching notes)
5. Have participants predict the number of seconds that they think it will take the thermometer to
“cool down” to room temperature.
7. What patterns do you see in the data? As time increases, the temperature decreases, then levels
off.
8. Sketch a scatter plot of the data collected on the grid below. Participants can also produce
this on their graphing calculator.
December 16, 2004. Ensuring Teacher Quality: Algebra II, produced by the Charles A. Dana Center at The University of Texas
at Austin for the Texas Higher Education Coordinating Board.
2
Algebra II: Strand 6. Exponentials and Logarithms; Topic 2. Applications; Task 6.2.3
9. What family of functions has the characteristic seen in the graph of the data? What is different
x
about the graph of the data and the usual ( b , where 0<b<1) parent function in this family.
Exponential functions. The graph of the data is approaching the room temperature, whereas
the typical parent function for an exponential of base b (0<b<1) approaches y = 0 .
10.
Adjust the list of data collected to create a scatter plot with a horizontal asymptote
at y = 0 . Use a graphing calculator to find an exponential regression equation of the form
y = ab x
that best fits the data.
Before entering a list in the calculator, we must subtract the room temperature from every
value in the table so that we can appropriately exponential regression to find the
exponential function that best fits the graph. The graph of this adjusted data will be a
vertical translation, making the graph have a horizontal asymptote at y = 0 .
Note: Recall that exponential regression uses the fact that x and log y are in a linear
relationship (that is “in the background” we find a regular linear regression of x to log y
and then use exponentials to get an equation that relates x and y. For example if the data is
truly modeled by
y = ab x
then we have that
log y = log(ab x ) = log a + log b x = log a + x log b . Thus log y would be a line
with vertical axis intercept log a and slope log b (note that since a and b are constants, so
are log a and log b). However, if that data is modeled by the equation y
= ab x + c , then
log y = log(ab x + c)
which is not linear (so we wouldn’t be able to perform linear
regression to determine the values of a, b, or c). But, if we shifted the data c units to the yaxis, then we could do the regression and just remember to shift the exponential equation
found.
Discuss why the data needs to be shifted down so that the horizontal asymptote is zero.
Next have them use their graphing calculators to find an exponential regression equation of
the form
y = abx
that best fits the data.
December 16, 2004. Ensuring Teacher Quality: Algebra II, produced by the Charles A. Dana Center at The University of Texas
at Austin for the Texas Higher Education Coordinating Board.
3
Algebra II: Strand 6. Exponentials and Logarithms; Topic 2. Applications; Task 6.2.3
11. Does this regression equation model the actual data? No, we must add the room temperature
back into the equation to obtain an accurate functional model of the data collected.
Part II:
This is an important part of the activity for it allows the presenter to tie the “Cooling Down”
experience to the transformations. The adding of 32 shifts the graph up 32 units, but the
multiplication by 1.8 stretches the graph. Hence, the two graphs will not run parallel to one
another.
Math notes
The data from this experiment provide an example of exponential decay and can be represented
x
by the function y = ab , where 0<b<1. The important point to stress is that 0<b<1 ensures that
we have exponential decay instead of exponential growth.
Teaching notes
Introduce the situation presented in the Task 6.2.3. Ask participants to discuss each of the
questions and encourage them to identify which graph represents each situation. Ask participants
to discuss which graph they believe best represents the cooling process for this situation. Wait to
answer this until after participants have collected the temperature data below.
Explain that, in this task, instead of letting the water cool and recording the resulting temperature,
we shorten the time interval necessary to collect the data by simulating water cooling. Thus, the
cooling process can be measured in seconds rather than in hours. To do this, we will place the
thermometer in a cup of hot water (≈180°F) for 1 minute to heat up. Then we will remove the
thermometer from the cup and let it cool to room temperature in the air. The thermometer cooling
in the air simulates the water cooling in the cup.
Some questions to ask at the beginning of the experiment:
• Describe the function we will be dealing with. The temperature reading of the thermometer
is a function of time.
• What is the independent variable? elapsed time
• What is the dependent variable? temperature reading
Also ask for a prediction: How long do you think it will take the thermometer to “cool down” to
room temperature?
December 16, 2004. Ensuring Teacher Quality: Algebra II, produced by the Charles A. Dana Center at The University of Texas
at Austin for the Texas Higher Education Coordinating Board.
4
Algebra II: Strand 6. Exponentials and Logarithms; Topic 2. Applications; Task 6.2.3
Demonstrate the procedure for the experiment below.
Have participants conduct the experiment in their groups.
Experiment Procedure without a data collection device. Have one participant from each
group handle the thermometer and call out the temperature. Another participant keeps track of
the elapsed time. The third participant records the data.
Although the independent variable is time, the participants will find it easier to call out
temperature decreases of 2° and record the elapsed time. The data recorder is responsible for
recording all data. All participants should be ready for quick readings initially. If they are
surprised by the quick drop in temperature readings, suggest that they start over.
Experiment Procedure using a data collection device.
Set up the experiment using a program which measure temperatures over an interval of time.
Have one participant from the group handle the temperature probe. Another participant operates
the graphing calculator. The third participant operates the electronic data collection device.
It is important to ask the following questions as students work on question 13.
• What patterns do you see in the data? As time increases, the temperature decreases, then
levels off.
• What information does the graph give you that was not as obvious from the table of values?
The rapid drop in temperature in the first several seconds.
! 1$
y=# &
" 2%
x
and y = 2
!x
•
How are the functions
related? They are the same.
•
How can an exponential function model a decaying (or decreasing) situation? In the
x
•
•
•
form y = ab , b must be between 0 and 1.
What function family has the characteristics seen in the graph of the data? exponential
functions.
What is the long run behavior of the data? The graph will level off at room temperature.
What is the long run behavior of an exponential function? As x gets large, the decreasing
exponential function has a horizontal asymptote. In this case, the exponential function
modeling the temperature drop has a horizontal asymptote at y = room temperature.
As participants attempt to answer question 11, listen to see if you hear them talking about
transformations or translations. Also, point out that in order to use the exponential regression
formula, we must account for the vertical translation in the graph (See the note in Question 10
above). Have them adjust the temperature data by subtracting the room temperature from each
value.
Using regression is tricky because with small data sets it may be easy to fit several different
curves to the data. However, in this case we “know” from Newton’s Law of Cooling that this
must be exponential. Thus, in question 11 we are not making a big leap in asking the participants
December 16, 2004. Ensuring Teacher Quality: Algebra II, produced by the Charles A. Dana Center at The University of Texas
at Austin for the Texas Higher Education Coordinating Board.
5
Algebra II: Strand 6. Exponentials and Logarithms; Topic 2. Applications; Task 6.2.3
to use the regression feature of their calculator. Ask the following questions as they work on
Question 11:
• What effect did subtracting the room temperature from the data points have? It allowed us to
use the exponential regression analysis by translating the graph to a general exponential
function.
• Why did we have to add the room temperature back into the regression equation? To make
our regression equation model the actual data.
• What would happen if we tried to find an exponential regression equation without
compensating for the room temperature shift? Have participants try it to see that the
regression equation would not fit the initial drop in temperature.
Now, return to the opening questions (1 & 2). They should conclude that the answer is Graph 2.
Refer back to the three graphs at the beginning of the task. Have participants reconcile their
original answer with their current answer, if they are different.
Technology notes
To use the CBL, press the APPS button on the calculator and then press 2.
The following screen will appear.
Press any key to continue and the following screen will appear.
Press 2 and you will see the following.
December 16, 2004. Ensuring Teacher Quality: Algebra II, produced by the Charles A. Dana Center at The University of Texas
at Austin for the Texas Higher Education Coordinating Board.
6
Algebra II: Strand 6. Exponentials and Logarithms; Topic 2. Applications; Task 6.2.3
Choose the PROBE you will use, etc. If you would like directions
on how to connect the CBL to the TI-83 Plus, turn the directions ON and then press GO.
The following sequence of screen shots is what you will see.
December 16, 2004. Ensuring Teacher Quality: Algebra II, produced by the Charles A. Dana Center at The University of Texas
at Austin for the Texas Higher Education Coordinating Board.
7
Algebra II: Strand 6. Exponentials and Logarithms; Topic 2. Applications; Task 6.2.3
TASK 6.2.3: EXPONENTIAL DECAY
Part I:
Consider a cup of hot water at approximately 180° F that is placed in a room where the
temperature is 74° F. Think about the cooling effect.
1. How will the water cool over time?
Graph 3
y
180
150
120
90
60
t
0.5
1.0
1.5
2.0
Time (hours)
2. Describe a situation implied by each graph.
December 16, 2004. Ensuring Teacher Quality: Algebra II, produced by the Charles A. Dana Center at The University of Texas
at Austin for the Texas Higher Education Coordinating Board.
8
Algebra II: Strand 6. Exponentials and Logarithms; Topic 2. Applications; Task 6.2.3
Use a thermometer:
We will simulate the water cooling down by recording the temperature data of a thermometer
cooling down.
Group Roles:
Temperature reader – handles the thermometer and calls out the temperature.
Timer – keeps track of the elapsed time.
Recorder – records the data.
3. Use the thermometer to determine the room temperature. ________
4. Pour hot, not boiling water (≈ 180° F) into a cup. Place the thermometer in the cup for about a
minute to heat up.
5. Predict the number of seconds that you think it will take your thermometer to “cool down” to
room temperature. _________
6. Remove the thermometer from the cup of hot water. (Make sure to move the thermometer
away from the steam of the cup.) Begin collecting data immediately. Although the
independent variable is time, you may find it easier to call out temperature decreases of 2°
and record the elapsed time. The data recorder is responsible for recording all data. Be
ready for quick readings initially.
Elapsed Time (seconds)
Temperature (°F or °C)
7. What patterns do you see in the data?
December 16, 2004. Ensuring Teacher Quality: Algebra II, produced by the Charles A. Dana Center at The University of Texas
at Austin for the Texas Higher Education Coordinating Board.
9
Algebra II: Strand 6. Exponentials and Logarithms; Topic 2. Applications; Task 6.2.3
8.
Sketch a scatter plot of the data collected on the grid below.
9. What family of functions has the characteristics seen in the graph of the data? What is
x
different about the graph of the data and the usual ( b where 0<b<1) parent function in this
family?
10.
Adjust the list of data collected to create a scatter plot with a horizontal asymptote
at y = 0 . Use a graphing calculator to find an exponential regression equation of the form
x
y = ab that best fits the data.
11. Does this regression equation model the actual data? If not, how can you adjust it so that it
does?
December 16, 2004. Ensuring Teacher Quality: Algebra II, produced by the Charles A. Dana Center at The University of Texas
at Austin for the Texas Higher Education Coordinating Board.
10
Algebra II: Strand 6. Exponentials and Logarithms; Topic 2. Applications; Task 6.2.3
Use a graphing calculator and an electronic data collection device with a temperature
probe.
We will simulate the water cooling down by recording the temperature data of a temperature
probe cooling down.
Group Roles:
Temperature reader - handles the temperature probe.
Calculator – handles the graphing calculator.
Recorder – handles the electronic data collection device.
1. Use the temperature probe to determine the room temperature. _______
2. Pour hot, not boiling water (≈ 180° F) into a cup. Place the temperature probe in the cup for
about a minute to heat up.
3. Predict and record the number of seconds that you think it will take your temperature probe
to “cool down” to room temperature.
4. Remove the temperature probe from the cup of hot water. (Make sure to move the
temperature probe away from the steam of the cup.) Begin looking at the data using the data
collection device.
5.
What patterns do you see in the data?
December 16, 2004. Ensuring Teacher Quality: Algebra II, produced by the Charles A. Dana Center at The University of Texas
at Austin for the Texas Higher Education Coordinating Board.
11
Algebra II: Strand 6. Exponentials and Logarithms; Topic 2. Applications; Task 6.2.3
Part II:
1. Convert the °C to °F by creating a new list L4 that is equal to 1.8 * L2 + 32. Record findings.
2. How are the two graphs related to each other. Why?
3. Create a scatter plot of L1, L4.
Select a window that will accommodate both the original scatter plot and the new scatter plot.
Sketch the plots below.
4. Compare and contrast the two graphs. Do the two graphs run parallel? If so, why? If not,
why not?
December 16, 2004. Ensuring Teacher Quality: Algebra II, produced by the Charles A. Dana Center at The University of Texas
at Austin for the Texas Higher Education Coordinating Board.