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Blessed be the God and Father of our Lord Jesus Christ, who has blessed us with all spiritual blessings in the heavenly places in Christ. Amen. - Ephesians 1:3

The Joy of a Teacher is the Success of his Students. - Samuel Dominic Chukwuemeka

Exponential Growth

Samdom For Peace

I greet you this day,
First: read the notes. Second: view the videos. Third: solve the questions/solved examples. Fourth: check your solutions with my thoroughly-explained solutions. Fifth: check your answers with the calculators as applicable.

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Samuel Dominic Chukwuemeka (Samdom For Peace) B.Eng., A.A.T, M.Ed., M.S

Objectives

Students will:

(1.) Discuss exponential growth.

(2.) Discuss real-world applications of exponential growth.

(3.) Solve problems involving exponential growth.



Formulas

(1.) $ N_f = N_ie^{kt} $


(2.) $N_f = N_i * 2$$\dfrac{t}{T_{double}}$


(3.) $ t = \dfrac{\ln{\left(\dfrac{N_f}{N_i}\right)}}{k} $


(4.) $ k = \dfrac{\ln{\left(\dfrac{N_f}{N_i}\right)}}{t} $


(5.) $ T_{double} = \dfrac{\ln 2}{k} $


(6.) $ k = \dfrac{\ln 2}{T_{double}} $


(7.) $ T_{double} = \dfrac{t\ln2}{\ln{\left(\dfrac{N_f}{N_i}\right)}} $


For Continuous Compound Interest:

(8.) $ A = Pe^{rt} $


(9.) $ P = \dfrac{A}{{e^{rt}}} $


(10.) $ t = \dfrac{\ln \left({\dfrac{A}{P}}\right)}{r} $


(11.) $ r = \dfrac{\ln \left({\dfrac{A}{P}}\right)}{t} $

Solved Examples

For ACT Students
The ACT is a timed exam...$60$ questions for $60$ minutes
This implies that you have to solve each question in one minute.
Some questions will typically take less than a minute a solve.
Some questions will typically take more than a minute to solve.
The goal is to maximize your time. You use the time saved on those questions you solved in less than a minute, to solve the questions that will take more than a minute.
So, you should try to solve each question correctly and timely.
So, it is not just solving a question correctly, but solving it correctly on time.
Please ensure you attempt all ACT questions.
There is no "negative" penalty for any wrong answer.

Solve each question.
Show all work.

(1.) In $2012$, the population of Samdom For Peace city was $6.03$ million.
The exponential growth rate was $1.84\%$ per year.
(a.) Determine the exponential growth function.
(b.) Estimate the population of the city in $2018$. Round to the nearest tenth as needed.
(c.) When will the population of the city be $10$ million? Round to the nearest tenth as needed.
(d.) Calculate the doubling time. Round to the nearest tenth as needed.


$ N_f = N_ie^{kt} \\[3ex] N_i = 6.03 \\[3ex] k = 1.84\% = 0.0184 \\[3ex] N_f = 6.03e^{0.0184t} \\[5ex] In\: 2018, \\[2ex] t = 2018 - 2012 = 6 \\[3ex] N_f = 6.03 * e^{0.0184 * 6} \\[3ex] N_f = 6.03 * e^{0.1104} \\[3ex] N_f = 6.03 * 1.116724671 \\[3ex] N_f = 6.733849766 \\[3ex] N_f = 6.7\:\: million \\[5ex] When\: N_f = 10, \\[2ex] t = \dfrac{\ln{\left(\dfrac{N_f}{N_i}\right)}}{k} \\[5ex] t = \dfrac{\ln{\left(\dfrac{10}{6.03}\right)}}{0.0184} \\[5ex] t = \dfrac{\ln{(1.658374793)}}{0.0184} \\[5ex] t = \dfrac{0.505838082}{0.0184} \\[5ex] t = 27.49120012 \\[2ex] t = 27.5\:\: years \\[5ex] T_{double} = \dfrac{\ln 2}{k} \\[5ex] T_{double} = \dfrac{0.693147181}{0.0184} \\[5ex] T_{double} = 37.67104242 \\[2ex] T_{double} = 37.7\:\: years $

(2.) A toy tractor sold for $$269$ in $1980$ and was sold again in $1990$ for $$497$.
Assume that the growth in the value of the collector's item was exponential.
Calculate the amount of time after which the value of the toy tractor will be $$2376$?
Round to two decimal places as needed.
Then, round to the nearest integer.
What year will that be? Interpret it.


In this case, we shall let $1980$ be the initial year.

From $1980$ to $1990$, the time, $t = 1990 - 1980 = 10\: years$

$N_i =$ $$269$

$N_f =$ $$497$

Let us calculate the growth constant.

$ k = \dfrac{\ln{\left(\dfrac{N_f}{N_i}\right)}}{t} \\[5ex] k = \dfrac{\ln{\left(\dfrac{497}{269}\right)}}{10} \\[5ex] k = \dfrac{\ln 1.847583643}{10} \\[5ex] k = \dfrac{0.613878646}{10} \\[5ex] k = 0.061387865 $
When $N_f =$ $$2376$

$k = 0.061387865$ (it is a constant)

$N_i =$ $$269$

$t = ?$

$ t = \dfrac{\ln{\left(\dfrac{N_f}{N_i}\right)}}{k} \\[5ex] t = \dfrac{\ln{\left(\dfrac{2376}{269}\right)}}{0.061387865} \\[5ex] t = \dfrac{\ln{8.832713755}}{0.061387865} \\[5ex] t = \dfrac{2.178462301}{0.061387865} \\[5ex] t = 35.48685560 \\[2ex] t = 35.49\: years $
$t = 35\: years$

The year will be $1980 + 35 = 2015$

This means that in the year, $2015$; the tractor is expected to sell for two thousand, three hundred and seventy six dollars.

(3.) ACT Observation of a certain bacteria colony has shown that its population of cells doubles every $3$ hours.
Given that the initial population of cells in this colony is about $8$ million, which of the following values, in millions, would be closest to the number of cells in the bacteria colony after $15$ hours?


We shall do this in three ways.
Choose any way you prefer.
ACT is a timed test...$60$ minutes to solve $60$ questions.
Ideally, each question should be solved in approximately a minute.
However, some questions take less than a minute to solve.
This means that the minutes saved from solving those questions should be used to solve questions that takes more than a minute.
The first method is recommended for the ACT
The first method is also recommended if you do not know the formula or if you were not given the formula.

First Method: Quantitative Reasoning

The population doubles every $3$ hours.
Initial population is $8$ million.
After the first $3$ hours, the population will be $8 * 2 = 16$ million
After the second $3$ hours ($6$ hours), the population will be $16 * 2 = 32$ million
After the third $3$ hours ($9$ hours), the population will be $32 * 2 = 64$ million
After the fourth $3$ hours ($12$ hours), the population will be $64 * 2 = 128$ million
After the fifth $3$ hours ($15$ hours), the population will be $128 * 2 = 256$ million

Second Method: Formula

$ T_{double} = 3 \:\:hours \\[3ex] N_i = 8 \:\:million \\[3ex] t = 15 \:\:hours \\[3ex] N_f = ? \\[3ex] N_f = N_ie^{kt} \\[3ex] k = \dfrac{\ln 2}{T_{double}} \\[5ex] k = \dfrac{\ln 2}{3} \\[5ex] kt = \dfrac{\ln 2}{3} * 15 = \ln 2 * 5 = 5\ln 2 \\[5ex] \implies N_f = 8 * e^{5\ln 2} \\[3ex] N_f = 8 * e^{\ln 2^5} \\[3ex] N_f = 8 * e^{\ln 32} \\[3ex] e^{\ln 32} = 32 ...Law\:\: 7...Log \\[3ex] N_f = 8 * 32 \\[3ex] N_f = 256 \:\:million \\[3ex] $ Third Method: Formula

$ T_{double} = 3 \:\:hours \\[3ex] N_i = 8 \:\:million \\[3ex] t = 15 \:\:hours \\[3ex] N_f = ? $ $N_f = N_i * 2$$\dfrac{t}{T_{double}}$

$ \dfrac{t}{T_{double}} = \dfrac{15}{3} = 5 \\[5ex] N_f = 8 * 2^5 \\[3ex] N_f = 8 * 32 \\[3ex] N_f = 256 \:\:million $

(4.) ACT The population of a particular town is modeled by the equation $P = 120,000(1.1)^t$ where $t$ is the number of years after January $1$, $2011$.
Based on the model, which of the following numbers is closest to the population of the town on January $1$, $2013$?

$ A.\:\: 132,000 \\[3ex] B.\:\: 145,000 \\[3ex] C.\:\: 160,000 \\[3ex] D.\:\: 264,000 \\[3ex] E.\:\: 396,000 $


$ P = 120,000(1.1)^t \\[3ex] t = 2013 - 2011 = 2 \\[3ex] P = 120,000(1.1)^2 \\[3ex] P = 145,200 \\[3ex] $ The closest is $145,000$

(5.) Assume the population of Love Thy Neighbor Country (wish it existed) has a population of four million, seven hundred and eighty three thousand people.
It has a land area of fourteen billion square yards.
It has an exponential growth rate of four and nine-tenths percent per year.
After how long will there be one person for every square yard of land?


$ Initial\:\:Population, N_i = 4783000 \\[3ex] Land\:\:Area = 14000000000\:\:square\:\:yards \\[3ex] Population\:\:Density = \dfrac{Initial\:\:Population}{Land\:\:Area} \\[5ex] Population\:\:Density = \dfrac{4783000}{14000000000} \\[5ex] Population\:\:Density = 0.000341642857\:\:people/square\:\:yard \\[3ex] But\:\:we\:\:need\:\:1\:person/square\:\:yard \\[3ex] This\:\:implies\:\:a\:\:Population\:\:Density\:\:of\:\:1\:\:person/square\:\:yard \\[3ex] This\:\:implies\:\:a\:\:population\:\:of\:\:14000000000 \\[3ex] \implies N_f = 14000000000 \\[3ex] When\:\:will\:\:that\:\:be\:\;means\:\:time? \\[3ex] time, t = ? \\[3ex] Growth\:\:rate, k = 4.9\% = \dfrac{4.9}{100} = 0.049 \\[5ex] t = \dfrac{\ln{\left(\dfrac{N_f}{N_i}\right)}}{k} \\[7ex] Numerator = \ln{\left(\dfrac{N_f}{N_i}\right)} = \ln{\left(\dfrac{14000000000}{4783000}\right)} \\[5ex] Numerator = \ln 2927.03324 \\[3ex] Numerator = 7.98174464 \\[3ex] Denominator = k = 0.049 \\[3ex] \rightarrow t = \dfrac{7.98174464}{0.049} \\[5ex] t = 162.892748\:\:years \\[3ex] t \approx 163\:\:years \\[3ex] $ It will take about a hundred and sixty three years for there to be one person per square yard of land.

Calculators for Exponential Growth

Given: growth rate

To Find: doubling time

in

Given: doubling time

To Find: growth rate

Required: initial amount, growth rate

Optional: time

To Find: other details

in

in "whatever unit"

Required: initial amount, growth rate

Optional: future amount

To Find: other details

in

Given: initial amount, time, doubling time

To Find: other details

in "whatever unit"

in "the same whatever unit"

Given: initial amount, time, future amount

To Find: other details

in "whatever unit"


References

Chukwuemeka, S.D (2016, April 30). Samuel Chukwuemeka Tutorials - Math, Science, and Technology. Retrieved from https://www.samuelchukwuemeka.com

Bittinger, M. L., Beecher, J. A., Ellenbogen, D. J., & Penna, J. A. (2017). Algebra and Trigonometry: Graphs and Models ($6^{th}$ ed.). Boston: Pearson.

Sullivan, M., & Sullivan, M. (2017). Algebra & Trigonometry ($7^{th}$ ed.). Boston: Pearson.

CrackACT. (n.d.). Retrieved from http://www.crackact.com/act-downloads/

VAST LEARNERS. (n.d). Retrieved from https://www.vastlearners.com/free-jamb-past-questions/