Math project

Florida Gateway CollegeCollege Algebra
MAC 1105
Description
This will be your main project for this College Algebra class. It is worth 3% of your
total course grade, included in the Instructor Discretion Category. You will write a
report explaining your work in detail. This project is mandatory for all students in this
course.
Objectives
In this project you will demonstrate your understanding of various topics related to
functions, and specifically exponential functions. The main objective is to create an
exponential model using data, and analyze it to predict how the given quantity will
behave. At the same time, you will work with and analyze concepts like:
• domain and range
• x- and y- intercepts
• finding the intersection of two graphs:
o by graphing
o by solving exponential equations
• growth rate of exponential functions
• function compositions
Grading
This project is worth 100 points and contributes 3% to your final course grade, included
in the Instructor Discretion Category. It will be graded based on the rubric that is
attached here. Please use this rubric as you progress to stay on track and make sure you
get a high score on your project. You can as much help as you need, but you will need
to document in your report who helped you at each step, so KEEP TRACK!
Writing the report
Use the attached blank form to write your answers. This is the report that you will
submit to your instructor for a grade.
Consider the following two scenarios:
• Situation #1: Initially, two people have heard a rumor. The number of people that
have heard the rumor doubles every 1 hour.
• Situation #2: Initially, three people have heard a rumor. The number of people
that have heard the rumor triples every 2 hours.
PART I. For each of these two situations:
1. Finding the values of the variables.
a. Create a table that gives the number of people N that have heard the rumor
after t hours, starting at t = 0.
2. Function behavior.
a. Is the behavior of N linear, polynomial or exponential?
b. How do you know?
c. What defines exponential behavior?
3. Constructing an equation (model) that gives the number of people N that
have heard the rumor after t number of hours.
a. Determine the pattern that relates N to t.
b. Write this pattern as a function 𝑁(𝑡) = 𝑁0 𝑏 𝑡 .
c. In your model (equation) 𝑁(𝑡) = 𝑁0 𝑏 𝑘𝑡 , explain the meaning of the
numbers b and N0.
d. Find the relevant domain and range of this function in the context of the
situation.
4. Changing the function to base e.
a. Using the inverse properties of exponential and logarithmic functions,
transform the equation to an equation with the natural base e.
b. In the model (equation) 𝑁(𝑡) = 𝑁0 𝑒 𝑘𝑡 , explain the meaning of the numbers
k and N0.
c. Find the value of the decay/growth constant k (to three decimal places) and
the initial value N0 in your model.
5. Applications of the model.
a. If there are 2,500 students at FGC, how long will it take (theoretically) for
everyone to have heard the rumor? (Round your answer to the nearest tenth
and provide the answer in hours and minutes.)
b. Find the current population of Lake City (cite source) and find how long it
will take (theoretically) for everyone in Lake City to have heard the rumor?
(Round your answer to the nearest tenth and provide the answer in hours
and minutes.)
c. Now do the same for the state of Florida. (Round your answer to the
nearest tenth and provide the answer in days and hours)
d. Finally, do the same calculation for the whole U.S. population (Round your
answer to the nearest tenth and provide the answer in days and hours).
PART II. Comparing the two situations:
1. In which situation will the rumor spread faster?
a. Use the growth constant to determine in which situation the rumor spreads
faster in the long run.
b. Using Desmos.com, graph each function on the same coordinate system
using a suitable domain and range.
c. Looking at the two graphs, how do you know in which situation the rumor
spreads faster?
2. When will the two situations give the same number of people who have heard
the rumor?
a. Find graphically the moment in time where both situations will give the
same number of people that have heard the rumor. Label this time as t = T.
b. Find the value of N (T).
c. Verify that both situations give the same number of people when t = T by
plugging the value of T in both models.
d. Describe what happens for times t < T and t > T.
3. Analytical calculation.
a. Now, find analytically the time T and verify that it agrees with the value
you found in problem 2a of section II.
THIS PAGE IS LEFT BLANK INTENTIONALLY
MAC 1105 – College Algebra
Class Project: Exponential Modeling
Name: ___________________________
Term/Year: ____________________
Section: __________
Professor: _____________________
PART I
1. Finding the values of the variables.
a. Create a table that gives the number of people N that have heard the rumor after t
hours, starting at t = 0.
Situation #1
Situation #2
t
(hours)
t
(hours)
N
(number of people
who have heard
the rumor)
0
1
N
(number of
people who have
heard the
rumor)
0
1
—
2
3
—
4
5
—
a. What is the behavior of N ? (circle one): Linear
Polynomial
2
3
4
5
2. Function behavior.
Exponential
b. How do you know?
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
c. What defines exponential behavior?
______________________________________________________________________
______________________________________________________________________
3. Constructing an equation (model) that gives the number of people N that have
heard the rumor after t number of hours.
a. Determine the pattern that relates N to t.
Situation #1
Situation #2
b. Write this pattern as a function N(t) = N0 bt.
Situation #1
Situation #2
c. In your model (equation) N(t)= N0 bkt, explain the meaning of the numbers b and
N0.
b = ______________________________________________________________
N0 = _____________________________________________________________
d. Find the relevant domain and range of this function in the context of the situation.
Situation #1 → Domain: _________________
Range: _________________
Situation #2 → Domain: _________________
Range: _________________
4. Changing the function to base e.
a. Using the inverse properties of exponential and logarithmic functions, transform
the equations in question 3b to an equation with the natural base e, that is
N(t)= N0 ekt.
Situation #1:
Situation #2:
b. Find the value of the decay/growth constant k (to three decimal places) and the
initial value N0 in your model.
Situation #1 → 𝑘: _____________________
𝑁0 : _____________________
Situation #2 → 𝑘: _____________________
𝑁0 : _____________________
c. In the model (equation) 𝑁(𝑡) = 𝑁0 𝑒 𝑘𝑡 , explain the meaning of the numbers k and
N0.
k = ______________________________________________________________
N0 = _____________________________________________________________
5. Applications of the model.
a. If there are 2,500 students at FGC, how long will it take (theoretically) for
everyone to have heard the rumor? (Round your answer to the nearest tenth and
provide the answer in hours and minutes.)
Situation #1 → Hours/Minutes: ____________
Situation #2 → Hours/Minutes: ___________
b. Find the current population of Lake City (cite source) and find how long it will
take (theoretically) for everyone in Lake City to have heard the rumor? (Round
your answer to the nearest tenth and provide the answer in hours and minutes.)
Population: ___________ Source: ____________________________________
Situation #1 → Hours/Minutes: ____________
Situation #2 → Hours/Minutes: ___________
c. Now do the same for the state of Florida. (Round your answer to the nearest tenth
and provide the answer in days and hours)
Population: ___________ Source: ____________________________________
Situation #1 → Hours/Minutes: ____________
Situation #2 → Hours/Minutes: ___________
d. Finally, do the same calculation for the whole U.S. population (Round your
answer to the nearest tenth and provide the answer in days and hours).
Population: ___________ Source: ____________________________________
Situation #1 → Hours/Minutes: ____________
Situation #2 → Hours/Minutes: ___________
PART II
1. In which situation will the rumor spread faster?
a. Use the growth constant to determine in which situation the rumor spreads faster
in the long run.
Circle one:
Situation #1
Situation #2
b. Using Desmos.com, graph each function on the same coordinate system using a
suitable domain and range. Sketch the results below.
c. Looking at the two graphs, how do you know in which situation the rumor
spreads faster?
_________________________________________________________________
_________________________________________________________________
2. When will the two situations give the same number of people who have heard
the rumor?
a. Find graphically the moment in time where both situations will give the same
number of people that have heard the rumor. Label this time as t = T. Put this on
the graph above.
b. From the graph, find the value of 𝑁(𝑇): N(T) = ________________________
c. Verify that both situations give the same number of people when t = T by
plugging the value of T in both models (equations).
𝑇 into Situation #1 Model
𝑇 into Situation #2 Model
d. Describe what happens for times t < T and t > T.
t < T:_____________________________________________________________ t > T:_____________________________________________________________
3. Analytical calculation.
a. Now, find analytically the time T and verify that it agrees with the value you
found in problem 2a of Part II.
Analytical Work (Show your work)
T from analytical calculation: __________________
T from the graph: __________________
Do they agree? (circle one):
YES
NO
LOG OF THE ASSISTANCE I RECEIVED ON THIS PROJECT
Part
Number Section
Person Who Helped Me
(I or II)
MAC1105: College Algebra
Exponential Model Project
Grading Rubric
Part–Number–Section
Student Name:
Semester and Year:
Professor:
Points Student
Possible Score
I–1–a
6
I–2–a
1
I–2–b
3
I–2–c
3
I–3–a
6
I–3–b
6
I–3–c
4
I–3–d
4
I–4–a
8
I–4–b
4
I–4–c
4
I–5–a
4
I–5–b
4
I–5–c
4
I–5–d
4
II – 1 – a
2
II – 1 – b
4
II – 1 – c
4
II – 2 – a
2
II – 2 – b
2
II – 2 – c
4
II – 2 – d
4
II – 3 – a
10
Help Log
3
TOTAL
100
Comments and Feedback
GLO LEVEL
Students will
accurately apply
mathematical
concepts.
Students will
apply
mathematical
reasoning to draw
valid conclusions.
Exceeds
Expectations
The student’s
work is clear and
accurate. Correct
responses were
given without
errors.
Meets
Expectations
The student’s
work is somewhat
accurate and
contains minimal
errors.
The student
accurately applied
mathematical
reasoning to draw
valid conclusions
without error.
The student
applied
mathematical
reasoning to draw
valid conclusions
with minor errors.
Needs
Improvement
The student has
only partially
explained all parts
of the solution, or
the responses
given are
incorrect.
The student drew
a valid conclusion
without
justification, or
the conclusion
was not valid.
Inadequate
The student did
not apply any
mathematical
concepts.
The student
neither applied
mathematical
reason nor drew
valid conclusions.