Page 1 of 6 MATH133 Unit 5: Exponential and Logarithmic Functions Individual Project Assignment: Version 2A Show all of your work details for these calculations. Please review this Web site to see how to type mathematics using the keyboard symbols. IMPORTANT: See Question 1 in Problem 2 below for special IP instructions. This is mandatory. Problem 1: Photic Zone Light entering water in a pond, lake, sea, or ocean will be absorbed or scattered by the particles in the water and its intensity, I, will be attenuated by the depth of the water, x, in feet. Marine life in these ponds, lakes, seas, and oceans depend on microscopic plant life that exists in the photic zone. The photic zone is from the surface of the water down to a depth in that particular body of water where only 1% of the surface light remains unabsorbed or not scattered. The equation that models this light intensity is the following: 𝐼 = 𝐼0𝑒−𝑘𝑥 In this exponential function, I0 is the intensity of the light at the surface of the water, k is a constant based on the absorbing or scattering materials in that body of water and is usually called the coefficient of extinction, e is the natural number 𝑒 ≅ 2.718282, and I is the light intensity at x feet below the surface of the water. 1. Choose a value of k between 0.025 and 0.095. 2. In a lake, the value of k has been determined to be the value that you chose above, which means that 100k% of the surface light is absorbed for every foot of depth. For example, if you chose 0.062, then 6.2% of the light would be absorbed for every foot of depth. What is the intensity of light at a depth of 10 feet if the surface intensity is I0 = 1,000 foot candles? (Correctly round your answer to one decimal place, and show the intermediate steps in your work.) http://www.purplemath.com/modules/mathtext.htm Page 2 of 6 3. What is the depth of the photic zone for this lake? (Hint: 𝐼 𝐼0 = 0.01, so 0.01 = 𝑒−0.062𝑥.) Solve this equation for x. Correctly round your answer to one decimal place and show the intermediate steps in your work. Problem 2: Compound Interest For discrete periods of time (once per year, twice per year, four times per year, 12 times per year, 365 times per year, etc.), the English terms we use to describe these, respectively, are annually, semiannually, quarterly, monthly, daily, etc. The formula for calculating the future amount when interest is compounded at discrete periods of time is 𝐴 = 𝑃 �1 + 𝑟 𝑛 � 𝑛𝑡 , where A is the amount you will have t years after the money is invested, P is the principal (the initial amount of money invested), r is the decimal equivalent of the annual interest rate (divide the interest rate by 100), and n is the number of times the interest is compounded in 1 year. For the compounding continuously situation, the formula is 𝐴 = 𝑃𝑒(𝑟)(𝑡), where A is the amount you will have after t years for principal, P, inv

1. Page 1 of 6
MATH133 Unit 5: Exponential and Logarithmic Functions
Individual Project Assignment: Version 2A
Show all of your work details for these calculations. Please
review this Web site to see how to
type mathematics using the keyboard symbols.
IMPORTANT: See Question 1 in Problem 2 below for special IP
instructions. This is
mandatory.
Problem 1: Photic Zone
Light entering water in a pond, lake, sea, or ocean will be
absorbed or scattered by the particles
in the water and its intensity, I, will be attenuated by the depth
of the water, x, in feet. Marine
life in these ponds, lakes, seas, and oceans depend on
microscopic plant life that exists in the
photic zone. The photic zone is from the surface of the water
down to a depth in that particular
body of water where only 1% of the surface light remains

2. unabsorbed or not scattered. The
equation that models this light intensity is the following:
� = �0�−��
In this exponential function, I0 is the intensity of the light at
the surface of the water, k is a
constant based on the absorbing or scattering materials in that
body of water and is usually called
the coefficient of extinction, e is the natural number � ≅
2.718282, and I is the light intensity at
x feet below the surface of the water.
1. Choose a value of k between 0.025 and 0.095.
2. In a lake, the value of k has been determined to be the value
that you chose above, which
means that 100k% of the surface light is absorbed for every foot
of depth. For example, if
you chose 0.062, then 6.2% of the light would be absorbed for
every foot of depth. What
is the intensity of light at a depth of 10 feet if the surface
intensity is I0 = 1,000 foot
candles? (Correctly round your answer to one decimal place,
and show the intermediate
steps in your work.)

3. http://www.purplemath.com/modules/mathtext.htm
Page 2 of 6
3. What is the depth of the photic zone for this lake? (Hint: �
�0
= 0.01, so 0.01 = �−0.062�.)
Solve this equation for x. Correctly round your answer to one
decimal place and show the
intermediate steps in your work.
Problem 2: Compound Interest
For discrete periods of time (once per year, twice per year, four
times per year, 12 times per year,
365 times per year, etc.), the English terms we use to describe
these, respectively, are annually,
semiannually, quarterly, monthly, daily, etc. The formula for
calculating the future amount when
interest is compounded at discrete periods of time is � = � �1
+ �
�
�
��
, where A is the amount
you will have t years after the money is invested, P is the

4. principal (the initial amount of money
invested), r is the decimal equivalent of the annual interest rate
(divide the interest rate by 100),
and n is the number of times the interest is compounded in 1
year.
For the compounding continuously situation, the formula is � =
��(�)(�), where A is the amount
you will have after t years for principal, P, invested at r decimal
equivalent annual interest rate
compounded continuously.
Based on the first letter of your last name, choose values from
the table below for P dollars and r
percent.
If your last name begins with
the letter
Choose an investment amount,
P, between
Choose an interest rate, r,
between
A–E $5,000–$5,700 9%–9.99%
F–I $5,800–$6,400 8%–8.99%

5. J–L $6,500–$7,100 7%–7.99%
M–O $7,200–$7,800 6%–6.99%
P–R $7,800–$8,500 5%–5.99%
S–T $8,600–$9,200 4%–4.99%
U–Z $9,300–$10,000 3%–3.99%
Page 3 of 6
Suppose that you invest P dollars at r% annual interest rate.
(Correctly round your answers to the
nearest whole penny (two decimal places), and show the
intermediate steps in all of these
calculations for full credit.)
1. Important: By Wednesday night at midnight, submit a Word
document containing
only your name and your chosen values from the table above for
P and r. Submit
this in the Unit 5 IP submissions area. This submitted Word
document will be used
to determine the Last Day of Attendance for government
reporting purposes.

6. 2. How much will you have in 8 years if the interest is
compounded quarterly?
3. How much will you have in 15 years if the interest is
compounded daily?
4. How much will you have in 12 years if the interest is
compounded continuously? Use
� ≅ 2.718282.
Problem 3: Newton’s Law of Cooling
According to Sir Isaac Newton’s Law of Cooling, the rate at
which an object cools is given by
the equation � = �� + (�0 − ��)�−��, where T is the
temperature of the object after t hours, T0
is the initial temperature of the object (when t = 0), Tm is the
temperature of the surrounding
medium, and k is a constant.
1. Suppose that a dessert at room temperature (T0 = 70°F) needs
to be frozen before it is
served. The dessert is placed in a freezer at Tm = 0°F. If the
value of the constant is k =
0.122, what will the temperature of the dessert be after 4 hours?
(Use � ≅ 2.718282,
correctly round your answer to two decimal places, and show
the intermediate steps in

7. your work.)
2. What do you think k in this formula represents?
3. Freezing is 32°F. How many hours will it take for this dessert
to freeze? (Correctly round
your answer to two decimal places, and show the intermediate
steps in your work.)
Problem 4: Medicare Expenditures
The following health care data represent health care
expenditures for years after 2000 in the
United States (U.S. Census Bureau, 2012):
Page 4 of 6
Actual Year Years After 2000 (x) Medicare Expenditures (in
billions of dollars)
2004 4 311.3
2006 6 403.1
2007 7 431.4
2008 8 465.7
2009 9 502.3

8. A natural logarithmic regression function model of the form,
�(�) = � + � ln(�), representing
these data can be found. These data can be closely modeled by
the following logarithmic
regression function:
�(�) = −9.5904 + 229.9582 ln(�)
1. Choose a value for x between 15 and 30 (it does not have to
be a whole number). Based
on this natural logarithmic function, what will be the
expenditure for health care in the
year represented by your chosen value of x? (Correctly round
your answer to one decimal
place, which is tenths of billions of dollars, and show the
intermediate steps in your work.
2. Based on this formula, in how many years after 2000 will the
health care expenditures be
$700 billion? (Correctly round your answer to one decimal
place, and show the
intermediate steps in your work.)
3. Using Excel or another graphing utility and the values from
the table above, draw the
graph of this function, �(�) = −9.5904 + 229.9582 ln(�). On
your graph does this

9. data seem to represent a natural logarithmic function? Explain
your answer. Is there
another function type that we have studied that seems to more
closely match the data?
Explain your answer.
4. In an English sentence, state the types of transformations of
the natural logarithmic
function, �(�) = ln(�), that will result in the following
function:
�(�) = −9.5904 + 229.9582 ln(�)
Problem 5: Richter Scale
Page 5 of 6
The Richter scale is a common logarithmic function (base 10)
based on a standard energy release
of �0 = 104.8 joules. The energy released by an earthquake, E
in joules, is then measured against
the standard by the formula, � ≅ 0.6667 log ��
�0
�, to get the Richter scale magnitude of the
earthquake, M.

10. 1. Based on this formula, complete the following table.
Correctly round your answer to one
decimal place, and show the intermediate steps in each of the
calculations. (Hint:
log(� × 10�) = log(�) + �; example log(5.0 × 105.2) = log(5)
+ 5.2 ≅ 0.69897 +
5.2 ≅ 5.89897 ≅ 5.9 rounded to one decimal place.) Please see
this Web site to for
help with exponent rules.
E
� =
�
��
�(�) ≅ �. �������(�)
0.5 x 106 0.5 x 101.2
1.0 x 108 1.0 x 103.2
1.5 x 1010 1.5 x 105.2
2.5 x 1012 2.5 x 107.2
1.6 x 1017 1.6 x 1012.2
Note: 1.6 x 1017 joules was the estimated energy released by
the San Francisco,

11. California earthquake on April 18, 1906 (Pidwirny, 2010).
2. According to the U. S. Geological Service (USGS), the
second strongest recorded
earthquake on Earth since 1900 occurred about 120 kilometers
southeast of Anchorage,
Alaska on March 27, 1964 (Historic Earthquakes, 2014). The
Richter magnitude of that
earthquake was registered at 9.2. What would be energy
released in joules of an
earthquake of magnitude 9.2? Correctly round your answer to
one decimal place, and
show the intermediate steps in your work. (Hint: Replace M(x)
by 9.2, and solve the
logarithmic equation for x; then multiply x by 104.8 to get the
value of E for this
magnitude.)
3. Which intellipath Learning Nodes helped you with this
assignment?
http://www.purplemath.com/modules/exponent.htm
Page 6 of 6

12. References
Exponents: Basic rules. (n.d.). Retrieved from the Purple Math
Web site:
http://www.purplemath.com/modules/exponent.htm
Formatting math as text. (n.d.). Retrieved from the Purple Math
Web site:
http://www.purplemath.com/modules/mathtext.htm
Historic earthquakes. (2014). Retrieved from the USGS Web
site:
http://earthquake.usgs.gov/earthquakes/states/events/1964_03_2
8.php
Pidwirny, M. (2010). Earthquake. Retrieved from the
Encyclopedia of Earth Web site:
http://www.eoearth.org/view/article/151858/
U.S. Census Bureau. (2012). Health and nutrition. Retrieved
from
http://www.census.gov/prod/2011pubs/12statab/health.pdf
http://www.purplemath.com/modules/mathtext.htm
http://earthquake.usgs.gov/earthquakes/states/events/1964_03_2
8.php
http://www.eoearth.org/view/article/151858/

13. Page 1 of 3
MATH133 Unit 4: Functions and Their Graphs
Individual Project Assignment: Version 2A
IMPORTANT: Please see Question 3 under Problem 2 for
special instructions for this
week’s IP assignment. This is mandatory.
Show all of your work details, explanations, and answers on the
Unit 4 IP Answer Form
provided.
Problem 1: Children’s Growth
A study of the data representing the approximate average
heights of children from birth to 12
years (144 months) has shown the following two equations. The
function is the radical
function representing the girls’ heights in inches after x months,
and the function is the radical
function representing the boys’ heights in inches after x months
( months).
( ) √
( ) √
1. Choose five different values of x between 0 and 144 months,

14. and calculate the values of each of
these functions for the chosen x values. Show all of your work
and display these calculated values
of ( ) and ( ) in “t-tables” in the Answer Form supplied.
2. Use these five different values of x and the corresponding
calculated values of both functions,
together with Excel or another graphing utility, to draw the
graphs of these two functions. These
graphs should be drawn on the same coordinate system so that
the two functions can be easily
compared. Insert those graphs into the Answer Form.
3. Set the two functions equal to each other, and solve the
resulting radical equation for x. This
value of x will be the age in months when boys and girls are the
same height. (Show all of the
steps for solving this radical equation on the Answer Form
provided.)
4. What is the height in inches when boys and girls (according
to these radical functions) are the
same height? (Show all of your work on the Answer Form
provided.)
5. Based on each of the two radical functions above, what is the
average change in height per

15. month for girls and the average change in height per month for
boys between the two values of
x (x = 30 months and x = 60 months)? (Show all of your work
on the Answer Form provided.)
Page 2 of 3
6. Describe the transformations of the radical function ( ) √
that will result in each of these
functions.
7. Which intellipath Learning Nodes helped you with this
problem?
Problem 2: Average Cost
Your company is making a product item. The fixed costs for
making this product are b, and the variable
costs are mx, where x is the number of items produced. The cost
function is the following linear function:
( )
The average cost is the total costs divided by the number of
items produced, which is a rational function,
as follows:
̅

16. 1. Based on the first letter of your last name, choose values for
m and b from the following tables
(Neither m nor b has to be a whole number):
First letter of your last name Possible values for m
A–F $10–$19
G–L $20–$29
M–R $30–$39
S–Z $40–$49
First letter of your last name Possible values for b
A–F $100–$149
G–L $150–$199
M–R $200–$299
S–Z $300–$399
2. Make up the type of company and a product that you think
fits the values of m and b that you
have chosen in Question 1, and briefly describe the company
and product. (There is no wrong

17. Page 3 of 3
answer except to not answer the question. Be creative in
developing your scenario, but do not
overdo it.)
3. Important: By Wednesday night at midnight, submit in a
Word document only your name
and your chosen values for m and b. Submit this in the Unit 4
submissions area. This
submitted Word document will be used to determine the Last
Day of Attendance for
government reporting purposes.
4. Choose five values of x < 50, and calculate the corresponding
values of ̅( ). Display these x and
̅( ) values in a t-table. (Show all of your work details for these
calculations. Please review this
Web site to see how to type mathematics using the keyboard
symbols.)
5. Using Excel or another graphing utility, draw the graph of
your average cost function, as follows:
̅

18. 6. What happens to your average cost rational function when x
gets very large? Explain your
answer.
7. How many items must be produced before the average cost is
1.5 times your chosen value of m?
(Show all of your work.)
8. Describe the transformations of the rational function ( )
that will result in your average
cost function. (Hint: What transformation types are used to get
from ( )
to ( )
(
)?)
9. Does your average cost function have a horizontal asymptote?

19. If so, what is that horizontal
asymptote equation? (Explain your answer.)
10. Which intellipath Learning Nodes helped you with this
problem?
Reference
Formatting math as text. (n.d.). Retrieved from the Purple Math
Web site:
http://www.purplemath.com/modules/mathtext.htm
http://www.purplemath.com/modules/mathtext.htm
http://www.purplemath.com/modules/mathtext.htm