LSP 120: Quantitative Reasoning and Technological Literacy

Section 903

Özlem Elgün

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.

HOW TO CALCULATE PERCENT CHANGEWe 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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x y Percent change 0 192 1 96 =(B3-B2)/B2 2 48 3 24

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

• 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.

x y Percent change 0 192 1 96 -50% 2 48 -50% 3 24 -50%

Which of the following are exponential?

x y 0 0 1 1 2 4 3 9

x y 0 3 1 5 2 9 3 13

x y 0 .5 5 1.5

10 4.5 15 13.5

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.

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

If a quantity changes by a fixed percentage,

it grows or decays exponentially.

How to increase/decrease a number by a percent

• There are 2 “ways” to do this:N = P + P * r

orN= P * (1 + r)

• According to the distributive property, these two formulas are the same. For the work we will be doing later in the quarter, 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.

Example1

Increase 50 by 10%N= 50 + 50 * .1 = 50 + 5 = 55

OR N = 50 * (1+.10) = 50 * 1.10 = 55

Reminder formulas:

Example 2• Sales tax is 9.75%. You buy an item for $37.00. What is

the final price of the article?

N = 37 + 37 * .0975 = 40.6075 or

N = 37 * (1+.0975) = 40.6075

• Since the answer is in dollars, round appropriately to 2 decimal places.

• The final price of the article is $40.61

Reminder formulas:

Example 3• In 1999, the number of crimes in Chicago was 231,265.

Between 1999 and 2000 the number of crimes decreased 5%. How many crimes were committed in 2000?

N = 231,265 - 231,265 * .05 = 219,701.75or

N = 231,265 * (1-.05) = 219,701.75

• Since the answer is number of crimes, round appropriately to the nearest whole number. The number of crimes in 2000 was 219,701.

Reminder formulas:

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).

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.

X minutes

Y population

0 100 1 =B2*(1+.05) 2 3 4 5

Reminder formula for calculating the new population after each period:

y=P*(1 + r) whereP= starting (reference, old) valuer= rate (percent change in decimals) y= the new value

• 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 = 1868. (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 1000. 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.

X minutes

Y population

0 100 1 =B2*(1+.05) 2 3 4 5

• Another example: Country A had population of 125 million in 1995. Its population was growing 2.1% a year. Country B had a population of 200 million in 1995. 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*(1+0.021) 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.

year A B 1995 125 200 1996 127.625 197.6 1997 130.3051 195.2288 1998 133.0415 192.8861 1999 135.8354 190.5714 2000 138.6879 188.2846 2001 141.6004 186.0251 2002 144.574 183.7928 2003 147.6101 181.5873 2004 150.7099 179.4083 2005 153.8748 177.2554 2006 157.1061 175.1283 2007 160.4054 173.0268 2008 163.7739 170.9505 2009 167.2131 168.8991 2010 170.7246 166.8723

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

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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year population (in millions) percent change (in decimal format) percent change (%)1995 1251996 127.625 (127.625-125)/125= .021 2.1%

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 1995. In the year 2010 its population was 170.7246 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= 170.7246 millionP= 125 million x= number of time periods from the initial value to the new value=2010-1995=15 yearsSolve for r!170.7246 = 125 *(1+r)15

170.7246/ 125 = (1+r)15

1.3657968 = (1+r)15

1.36579681/15 = (1+r)1.020999994 =1+r1.020999994 -1 = r0.020999994 = r0.021 = r 2.1%

The population grew 2.1 percent every year in country A from 1995 to 2010.

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 1995. 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=1995-1990=5 yearsr=2.1% (0.021 in decimals)Solve for P!

125 = P*(1+0.021)5

125 = P*(1.021)15

125= P*1.109503586125/1.109503586= P112.6629977=P

In year 1990, there were 112.66 million people in country A.

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 2000. It grew 15% in 10 years. What was the population in 2010?