1. LSP 120: Quantitative Reasoning and Technological Literacy
Topic 2: Exponential Models
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Exponential Change
Like linear relationships, exponential relationships describe a specific pattern of change between a dependent and independent variable
Recognizing and modeling this pattern allows us to make predictions about future values based on an exponential rate of change.
While in linear relationships, y changes by a fixed absolute amount for each change in x (such as adding 3 each time), in exponential relationships, y changes by a fixed relative amount for each change in x (such as adding 3% each time).
An exponential relationship is one in which for a fixed change in x, there is a fixed percent change in y.
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HOW TO CALCULATE PERCENT CHANGE
We can use Excel to determine if a relationship is exponential by filling the neighboring column with the percent change from one Y to the next. As with linear, do not put this formula in the cell next to the first pair of numbers but in the cell next to the second.
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4. How to Format Cells in Excel
Right click on the cell you want to format
Click on ‘Format Cells…’
Select the ‘Number’ Tab
Choose the appropriate format you want
i.e. If the cell contains a number, select ‘Number’
Display at least two decimal points
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You can display percent change in decimal or percent format. For descriptive or visual representations you want to use the percent format. For calculations you want to use the decimal format.
Above, I explain how to change the format of your cells. An easy way to format your cells to display percentages is to click on the ‘%’ icon on Excel.
If the column is constant, then the relationship is exponential. So this function is exponential.
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Which are exponential?
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8. Exponential function equation
As with linear, there is a general equation for exponential functions. The equation for an exponential relationship is:
y = P*(1+r)x
P = initial value (value of y when x = 0),
r is the percent change (written as a decimal),
x is the input variable (usually units of time),
Y is the output variable (i.e. population)
The equation for the above example would be
y = 192 * (1- 0.5)x Or
y = 192 * 0.5x.
We can use this equation to find values for y if given an x value.
Write the exponential equation for other examples.
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Why are exponential relationships important? Where do we encounter them?
Populations tend to growth exponentially not linearly
When an object cools (e.g., a pot of soup on the dinner table), the temperature decreases exponentially toward the ambient temperature (the surrounding temperature)
Radioactive substances decay exponentially
Bacteria populations grow exponentially
Money in a savings account with at a fixed rate of interest increases exponentially
Viruses and even rumors tend to spread exponentially through a population (at first)
Anything that doubles, triples, halves over a certain amount of time
Anything that increases or decreases by a percent
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If a quantity changes by a fixed percentage, it grows or decays exponentially.
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11. How to increase/decrease a number by a percent
There are 2 “ways” to do this:
N = P + P * r or
N= P * (1 + r)
According to the distributive property, these two formulas are the same. For later work, the second version is preferred.
Similarly to the formula above, N is the ending value (new value), P is the starting value (initial, reference, old value) and r is the percent (written in decimal form). To write a percent in decimal form, move the decimal 2 places to the left. Remember that if there is a percent decrease, you will be subtracting instead of adding.
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Example1
Reminder formulas:
Increase 50 by 10%
N= * .1 =
= 55 OR
N = 50 * (1+.10)
= 50 * 1.10
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Example 2
Reminder formulas:
Sales tax is 9.75%. You buy an item for $ What is the final price of the article? 
N = * = or
N = 37 * ( ) =
Since the answer is in dollars, round appropriately to 2 decimal places.
The final price of the article is $40.61
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Example 3
Reminder formulas:
In 2008, the number of crimes in Chicago was 168,993. Between 2008 and 2009 the number of crimes decreased 8.7%. How many crimes were committed in 2009?
N = * .087 = 154,290.6 or
N = * (1-.087) = 154,290.6
Round to nearest whole number
154,291 crimes
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Exponential growth or decay is increasing or decreasing by same percent over and over
If a quantity P is growing by r % each year, after one year there will be
P*(1 + r)
So, P has been multiplied by the quantity 1 + r.  
If  P*(1 + r) is in turn increased by r percent, it will be multiplied by (1 + r) again. 
So after two years, P has become:
P*(1+r)*(1+r) = P*(1 + r)2
So after 3 years, you have P*(1+r)3, and so on. Each year, the exponent increases by one since you are multiplying what you had previously by another (1+r).
If a quantity P is decreasing by r%, then by the same logic, the formula is P*(1  r).
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Example: A bacteria population is at 100 and is growing by 5% per minute. 2 questions: 1. How many bacteria cells are present after one hour? 2. How many minutes will it take for there to be 1000 cells.
If the population is growing at 5% a minutes, this means it is being multiplied by 1.05 each minute. Using Excel, we can set up a table to calculate the population at each minute. Note that the time column begins with 0.
The formula in the second cell of the column is the cell above it multiplied by 1 + the percent written as a decimal. (Note that you do not enter the exponent in the formula) Filling the column will give us the population at each minute.
Reminder formula for calculating the new population after each period:
y=P*(1 + r)
where
P= starting (reference, old) value
r= rate (percent change in decimals)
y= the new value
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If you wanted to calculate the population after one hour (60 minutes), you could drag the columns of the excel table down to 60 or you could use the following formula which I refer to as the “by hand” formula. Note that here because you are doing the calculation in one step you do use the exponent.
Y = P *(1+r)x
Here x is the number of minutes.
Therefore, the population after 60 minutes = y = 100*(1+.05)60 = (To enter an exponent in Excel use the ^ key which is above the number 6.) Remember to round your answer appropriately. For this example, we should round to the nearest whole number.
If you wanted to know how long it would take for the population to reach 1000, you could set up the table in Excel and drag the columns down until the population (the Y value) reaches Doing so, you should find that the population will reach 1000 after 48 minutes. We will discuss a more mathematical and exact way to solve this problem in the near future.
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Another example: Country A had population of 125 million in Its population was growing 2.1% a year. Country B had a population of 200 million in Its population was decreasing 1.2% a year. What are the populations of the countries this year? Has the population of country A surpassed the population of B? If not, in what year will the population of country A surpassed the population of B?
Create your own Excel table. The formula in cell B3 is =B2*( ) and in C3 is =B2*(1-.012). Then each column is filled. Since the A’s population was not larger than B’s in 2006, the columns needed to be extended down farther.
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19. Solving Exponential Equations
Remember that exponential equations are in the form:
y = P(1+r)x
P is the initial (reference, old) value
r is the rate, a.k.a. percent change (and it can be either positive or negative)
x is time (years, minutes, hours, seconds decades etc…)
Y is the new value
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20. Solving for rate(percent change) and the initial value
Solving for the rate(percent change):
We already know that we solve for percent change between two values that are one time period apart:
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21. Solving for the rate(percent change)
What if we had to solve for the rate (percent change) but the new value is not one time period but several time periods ahead? How do we solve for the rate then? Example: The population of country A was 125 million in In the year 2010 its population was million people. If we assume that the percent change (rate of population growth) was constant, at what rate did the population grew? reminder our formula is  y = P*(1+r)x In this case: y= million P= 125 million x= number of time periods from the initial value to the new value= =15 years Solve for r! = 125 *(1+r) / 125 = (1+r) = (1+r) /15 = (1+r) =1+r = r = r = r  2.1% The population grew 2.1 percent every year in country A from 1995 to 2010.
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22. Solving for the initial value (a.k.a. old value, reference value)
What if we had to solve for the initial value? Example: The population of a country was 125 million in If we knew that the percent change (rate of population growth) was constant, and grew at a rate of 2.1 percent every year, what was its population in year 1990? reminder our formula is  y = P*(1+r)x In this case: y= 125 million x= number of time periods from the initial value to the new value= =5 years r=2.1% (0.021 in decimals) Solve for P and verify with your spreadsheet that
125 = P*( )5
125 = P*(1.021)5
125= P*
125/ = P =P
In year 1990, there were million people in country A.
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Important reminder
You use y = P*(1+r)x formula if r is increasing yearly/monthly etc. and you are asked to calculate a value over a number of time periods.
Example: the population is growing at 0.01% yearly. The population is 1 million now; what will it be in 10 years?
You use y = P*(1+r) formula (without x) if you are given an r value that does not grow annually/monthly etc. but represents overall percent change across time.
Example: The population is was 1 million in It grew 15% in 10 years. What was the population in 2010?
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24. Solving for time (x) using Excel
Example: The population of Bengal tigers was 4700 in If we assume that the rate of population growth will remain constant, and will be at a rate of 2.3 percent every year, how many years have to pass for Bengal tiger population to reach 10,000?
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26. Solving for time (using logarithms)
To solve for time, you can get an approximation by using Excel. To solve an exponential equation algebraically for time, you must use logarithms.
There are many properties associated with logarithms. We will focus on the following property:
log ax = x * log a for a>0
This property is used to solve for the variable x (usually time), where x is the exponent.
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Solving time (x) with logarithms:
Example: The population of Bengal tigers was 4700 in If we assume that the rate of population growth will remain constant, and will be at a rate of 2.3 percent every year, how many years have to pass for Bengal tiger population to reach 10,000?
Start with : Y= P * (1 + r)X.
Fill the variables that you know. To use logarithms, x (time) must be your “unknown” quantity.
 y= 10000
P= 4700
R=0.023
Solve for x!
The equation for this situation is: = 4700 * ( )X
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Solving time (x) with logarithms: Continued…. We need to solve for x: Step 1: divide both sides by /4700 = ( )X = ( )X USE THE LOG property you learned earlier  log ax = x * log a for a>0 Step 2: take the log of both sides log( ) = log ( )X Step 3: bring the x down in front log( ) = x * log ( ) Step 4: divide both sides by log (1+.023) log( ) /log(1+.023) = x to get Step 5: Write out your answer in words: If we assume that the rate of population growth will remain constant, and will be at a rate of 2.3 percent every year 33.2 years have to pass for Bengal tiger population to reach 10,000.
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29. What is Radioactivity?
Emission of energy (or particles) from the nuclei of atoms
Atoms like to have the same # of protons and neutrons
Like to be stable
Unstable atoms are radioactive
Throw off parts to become stable

30. Good or Bad Good Bad Used in medicine to treat heart disease, cancer
Kills bacteria like salmonella
Carbon dating
Bad
Uncontrolled nuclear chain reactions can cause major damage
Like Chernobyl
Can cause burns to cell mutations

31. Application of Exponential Models: Carbon Dating
A radioisotope is an atom with an unstable nucleus, which is a nucleus characterized by excess energy which is available to be imparted either to a newly-created radiation particle within the nucleus, or else to an atomic electron. The radioisotope, in this process, undergoes radioactive decay, and emits a gamma ray(s) and/or subatomic particles. These particles constitute ionizing radiation. Radioisotopes may occur naturally, but can also be artificially produced.
Radiocarbon dating, or carbon dating, is a radiometric dating method that uses the naturally occurring radioisotope carbon-14 (14C) to determine the age of carbonaceous materials up to about 58,000 to 62,000 years
One of the most frequent uses of radiocarbon dating is to estimate the age of organic remains from archaeological sites.
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The Dead Sea Scrolls are a collection of 972 documents, including texts from the Hebrew Bible, discovered between 1946 and 1956 in eleven caves in and around the ruins of the ancient settlement of Khirbet Qumran on the northwest shore of the Dead Sea in the West Bank.
We date the Dead Sea Scrolls which have about 78% of the normally occurring amount of Carbon 14 in them.  Carbon 14 decays at a rate of about 1.202% per 100 years. I make a table of the form.
Using excel and extending the table we find that the Dead Sea Scrolls would date from between 2100 to 2000 years ago.
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34. Use logarithms to solve for time
When were the dead sea scrolls created?
78 = 100*( )X
.78=( ) X
Log (.78)= x*log ( )
= x*( )
/ =x
20.546=x
Since x is in units of 100 years
Dead Sea Scrolls date back years
(Current estimates are that a 95% confidence interval for their date is 150 BC to 5 BC)
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Activity 3, Question 1: Beryllium-11 is a radioactive isotope of the alkaline metal Beryllium.  Beryllium-11 decays at a rate of 4.9% every second. 
a) Assuming you started with 100%, what percent of the beryllium-11 would be remaining after 10 seconds? Either copy and paste the table or show the equation used to answer the question. y = 100*( )10 =100*(0.951) 10 Y= % Beryllium-11 remains after 10 seconds.
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Activity 3, Question 1 Beryllium-11 is a radioactive isotope of the alkaline metal Beryllium.  Beryllium-11 decays at a rate of 4.9% every second. 
b) How long would it take for half of the beryllium-11 to decay? This time is called the half life. (Use the "solve using logs" process to answer the question) Show your work. 50 =100*( ) X .50 =(0.951) X Log (.50) = x*log (0. 951) = x*( ) / =x =x It would take seconds for Beryllium-11 to reach its half life.
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