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.