1
Algebra II: Strand 6. Exponentials and Logarithms; Topic 2. Applications; Task 6.2.1
TASK 6.2.1: E. COLI GROWTH RATE
Solutions
Elapsed time
(min)
20
50
80
110
130
165
180
200
230
Temp = 30° C
Klett value
5
6
7
17
22
30
35
40
57
LB Cultures
Temp = 37° C
Number of
Klett
Number of
cells/ml
value
cells/ml
2.50 x 107
3.00 x 107
3.50 x 107
8.50 x 108
1.10 x 108
1.50 x 108
1.75 x 108
2.00 x 108
2.85 x 108
2
6
11
27
37
62
75
87
106
Superbroth Cultures
Temp = 30° C
Temp = 37° C
Klett
Number of
Klett
Number of
value
cells/ml
value
cells/ml
1.00 x 107
3.00 x 107
5.50 x 107
1.35 x 108
1.85 x 108
5.10 x 108
3.75 x 108
4.35 x 108
5.30 x 108
2
7
7
10
16
20
27
38
53
LB Culture
(Standard Nutrient Broth)
1.00 x 107
3.50 x 107
3.50 x 107
5.00 x 107
8.00 x 107
1.00 x 108
1.35 x 108
1.90 x 108
2.65 x 108
2
7
15
30
53
87
125
163
188
1.00 x 107
3.50 x 107
7.50 x 107
1.50 x 108
2.65 x 108
4.35 x 108
6.25 x 108
8.15 x 108
9.40 x 108
Superbroth
(Enriched Nutrient Broth)
Klett 1 = 5 x 106 cells/ml Klett 10 = 5 x 107 cells/ml Klett 100 = 5 x 108 cells/ml
1. Set up a scatterplot for the E. coli Growth Data using Klett values: A possible calculator
window [-10, 250] [-10, 70].
a.
LB broth at 30°C versus elapsed time
b.
LB broth at 37°C versus elapsed time
c.
Superbroth at 30°C versus elapsed time
d.
Superbroth at 37°C versus elapsed time.
2. Sketch your plot(s) on large grid paper and on the grid on the next page. Label your graphs.
Plots will vary according to scales used.
3. Find a model for your data. Sketch the model for each plot on the large grid paper and on the
next page. Label the model with the equation.
b.
c.
d.
x
(
))
Regression equation ( y = 2.308 (1.0189 ) )
Regression equation ( y = 2.246 (1.0141) )
Regression equation ( y = 2.246 (1.0141) )
a. Regression equation ( y
= 3.558 1.0125
x
x
x
4. Pose two good questions that can be answered using your graphs, tables, or models and
answer your questions. What interpretations or predictions can you make using your graphs,
tables, or data? Answers vary.
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 6.2.1
Math notes
It is important to emphasize the characteristics present in the scatter plots of data that may
indicate the best model is exponential. In particular, although many models look as though they
grow exponentially, in reality they may level off. Logistic curves may represent a more realistic
model.
Teaching notes
Task 6.2.1 should be preceded with the following discussion:
• How might a scientist measure the number of bacteria in a sample?
To monitor population growth of bacteria, biologists could examine a small sample under
a powerful microscope and count the population number at different times. This
approach, however, takes a great deal of time and is not particularly accurate.
A more expedient and accurate approach takes advantage of the fact that as bacteria
multiply, they increasingly cloud the solution in which they are growing. The cloudiness
occurs as a result of the bacteria absorbing (or scattering light). The more bacteria, the
more that light is absorbed (or scattered), and hence the cloudier the solution becomes.
An instrument, known as a spectrophotometer, measures the amount of light that is
absorbed by a solution. Absorbance (or optical density) is directly related to the number
of light-absorbing cells. The optical density is proportional to the number of cells per
unit of solution. This implies that the optical density (concentration of bacteria in a
solution) is equal to a constant times the number of cells per unit.
In an experiment, measurements were taken with a Klett-Summerson colorimeter, which
is a simple type of spectrophotometer that measures the optical density of a culture.
Use the transparency of the table in Task 6.2.1 to discuss the experiment as follows.
Two different nutrient solutions were used to grow E. coli bacteria cultures:
1. LB, a standard nutrient broth and
2. Superbroth, an enriched nutrient broth.
Four flasks, two containing 30 milliliters of LB solution and two containing 30 milliliters
of Superbroth solution, were each inoculated with 0.3 milliliters of an overnight culture
of E. coli bacteria.
Two temperature conditions were used:
1. 30˚ C (between room temperature and body temperature)
2. 37˚ C (body temperature).
The data table gives Klett readings and number of cells per milliliter as functions of
elapsed time in minutes for the following 4 situations:
1. LB Broth at 30˚ C
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 6.2.1
2. LB Broth at 37˚ C
3. Superbroth at 30˚ C
4. Superbroth at 37˚ C.
A Klett reading of 1 corresponds to 5 × 106 cells/milliliter.
A Klett reading of 10 corresponds to 5 × 107 cells/milliliter.
A Klett reading of 100 corresponds to 5 × 108 cells/milliliter.
Ask participants to complete Task 6.2.1 by investigating the growth patterns in each of the four
given situations using graphing utilities and determining mathematical models for each situation.
To save time, assign each group one situation and then have them present their findings to the
whole group.
1.
What is the constant of proportionality? 5 × 106
2.
If the optical density were 20, then how many bacteria would there be per milliliter? 20
× 5 × 106 cells/ml or 108 cells/ml
3.
If the optical density were 57, then how many bacteria would there be per milliliter? 57
× 5 × 106 cells/ml or 2.85 × 108 cells/ml
4.
What type of function relationship exists between Klett readings and the corresponding
number of bacteria per milliliter? linear or proportional
Each group should:
1. use a graphing calculator to make a scatter plot of the E. coli growth data assigned to
their group. [independent variable = time, dependent variable = Klett value]
2. choose a viewing window that will display complete plots.
3. make a hard copy of the plot on large grid paper and on the Activity Sheet.
4. find an exponential model for the data.
5. graph the exponential function on the hard copy.
6. repeat the process for one of the other situations if time permits.
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 6.2.1
TASK 6.2.1: E. COLI GROWTH RATE
Elapsed
time
(min)
20
50
80
110
130
165
180
200
230
LB Cultures
Temp = 30° C
Temp = 37° C
Klett
Number
Klett
Number
value
of
value
of
cells/ml
cells/ml
5
6
7
17
22
30
35
40
57
2.50 x 107
3.00 x 107
3.50 x 107
8.50 x 108
1.10 x 108
1.50 x 108
1.75 x 108
2.00 x 108
2.85 x 108
2
6
11
27
37
62
75
87
106
1.00 x 107
3.00 x 107
5.50 x 107
1.35 x 108
1.85 x 108
5.10 x 108
3.75 x 108
4.35 x 108
5.30 x 108
LB Culture
(Standard Nutrient Broth)
Superbroth Cultures
Temp = 30° C
Temp = 37° C
Klett
Number
Klett
Number
value
of
value
of
cells/ml
cells/ml
2
7
7
10
16
20
27
38
53
1.00 x 107
3.50 x 107
3.50 x 107
5.00 x 107
8.00 x 107
1.00 x 108
1.35 x 108
1.90 x 108
2.65 x 108
2
7
15
30
53
87
125
163
188
1.00 x 107
3.50 x 107
7.50 x 107
1.50 x 108
2.65 x 108
4.35 x 108
6.25 x 108
8.15 x 108
9.40 x 108
Superbroth
(Enriched Nutrient Broth)
Klett 1 = 5 x 106 cells/ml Klett 10 = 5 x 107 cells/ml Klett 100 = 5 x 108 cells/ml
1. Set up a scatterplot for the E. coli Growth Data using Klett values:
a. LB broth at 30°C versus elapsed time
b. LB broth at 37°C versus elapsed time
c. Superbroth at 30°C versus elapsed time
d. Superbroth at 37°C versus elapsed time.
2. Sketch your plot(s) on large grid paper and on the grid on the next page. Label your graphs.
3. Find a model for your data. Sketch the model for each plot on the large grid paper and on the
next page. Label the model with the equation.
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 6.2.1
4. Pose two good questions that can be answered using your graphs, tables, or models and answer
your questions. What interpretations or predictions can you make using your graphs, tables, or
data?
5. Graph the data and a model for the data. Label. Be creative.
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.