Transcript Exponential Equations

Finding an Exponential Regression
Use the data in the
program file COOL
to find an exponential
model.
© 2002 Jerel L. Welker, Lincoln High School
Permission is granted to copy and distribute
with copyright notation attached.
Finding an exponential model
The data was collected from a
hot water bath as it cooled to
an ambient temperature of
27.7 degrees Celcius. One
sample was taken every 60
seconds for an hour.
Pressing STAT - EDIT, one
can see the data is in
seconds.
For easier analysis, convert
the seconds to minutes.
y = k + a*rx
Co
Time sec
Finding an exponential model
To convert time from
seconds to minutes, divide
by 60. Place the cursor on
L1. Enter the formula L1/60
and press ENTER.
The data will reflect the
conversion and appear
similar to the table at the
right.
To redraw the graph with the
new coordinates, press
ZOOMSTAT (Zoom - 9)
y = k + a*rx
Finding an exponential model
The graph will now be
degrees Celcius vs Time in
minutes.
y = k + a*rx
Co
Time min
Finding an exponential model
y = k + a*rx
Solving Algebraically: Step 1 - Find k
The exponential model y = k + a*rx will approach y = k
as x approaches infinity if 0 < r < 1. Since the ambient
room temperature was reported to be 27.7o C, the water
will approach room temperature and k = 27.7o C
Finding an exponential model
y = k + a*rx
Solving Algebraically: Step 2 - Find a
If x=0, the model y = k + a*rx is y = k + a since r0 = 1.
Tracing the graph
where x=0, finds the
y-value to be 88.15o
Solving:
88.15 = 27.7 + a
a = 60.45o C
Finding an exponential model
y = k + a*rx
Solving Algebraically: Step 3 - Find r
Choose another ordered pair (x, y) and solve for ‘r’.
Trace to find another
point on the graph to
substitute.
y = k + a*rx
Finding an exponential model
Solving Algebraically: Step 3 - Find ‘r’ (con’t)
Substituting (50, 43.92) and solving for ‘r’:
43.92o =27.7o + 60.45o * r50
43.92o - 27.7o
60.45o
(
= r50
(43.92o - 27.7o)
60.45o
)
1
50
=r
r = 0.9740
Finding an exponential model
Solving Algebraically: Step 4 - Graph
Graph y = 27.7 + 60.45 * .9740x
y = k + a*rx
Finding an exponential model
y = k + a*rx
Using a regression: Step 1 - Shift the data to approach zero
The TI-83 exponential regression is in the form of y = a*bx
which assumes the model will approach 0 as x approaches
infinity. Since our data approaches 27.7, we must shift the
data down so that it will approach 0. After the regression is
complete, the model will be shifted back up the 27.7
degrees.
Press STAT - EDIT and
place the cursor on L3. Enter
the formula L2 - 27.7 and
press ENTER. L3 will be
the temperature reduced to
approach 0.
Finding an exponential model
y = k + a*rx
Using a regression: Step 1 - Shift the data to approach zero
The data in L3 will appear
as:
Using a regression: Step 2 - Fit the regression.
Press STAT - CALC and choose
ExpReg.
Finding an exponential model
y = k + a*rx
Using a regression: Step 2 - Fit the regression.
The y-values are in L3.
Set the regression for the
x-values in L1 and the yvalues in L3.
The resulting equation was
stored in Y2
The resulting regression
must be shifted up by 27.7
degrees to fit the original
data.
(con’t)
Finding an exponential model
y = k + a*rx
Using a regression: Step 3 - Graph
Graphing the regression
determined in Step 2, one
finds the graph at the
right.
Using a regression: Step 4 - Write a mathematical model
ToC = 27.7oC + 54.24oC * .9758 Tmin
Using a regression: Step 4 - Explain the mathematical model
The ambient temperature is 27.7o. The initial
temperature is 54.24o greater than the ambient
temperature and retains 97.58% of its temperature each
minute or it decreases its temperature 2.42% each minute.