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9TP6Exponential Modeling
A linear relationship is one in which there is a fixed rate of change (slope). In an exponential relationship as one variable increases by 1, the other increases or decreases by a fixed percent.
For example: Which of the following are exponential?
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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.
EMBED Equation.3
xyPercent change 0192196=(B3B2)/B2248324If doing this calculation on a calculator, you will need to multiply by 100 or move the decimal 2 places to the right to convert to a percent. In Excel, just click on the % icon on the toolbar.
xyPercent change 019219650%24850%32450%If the column is constant, then the relationship is exponential. So this function is exponential.
As with linear, there is a general equation for exponential functions. The equation for an exponential relationship is y= P*(1+r)x,
where P is the starting value (value of y when x = 0), r is the percent change (written as a decimal), and x is the input variable (usually time).
The equation for the above example would be y = 192 * (1.5)x or y = 192 * .5x.
We can use this equation to find values for y if given an x value.
Try the 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.
Lets look at how to increase or decrease a number by a percent. There are 2 ways to do this:
N = P + P * r
N= 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, P is the starting 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.
How to increase or decrease a number by a percent:
Examples:
Increase 50 by 10%
N= 50 + 50 * .1 = 50 + 5 = 55 OR
N = 50 * (1+.10) = 50 * 1.10 = 55
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
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 = 219,701.75
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 219,702.
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.
X
minutesY
population01001=B2*(1+.05)2345
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.
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 = 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.
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 As population was not larger than Bs in 2006, the columns needed to be extended down farther.
yearAB19951252001996127.625197.61997130.3051195.22881998133.0415192.88611999135.8354190.57142000138.6879188.28462001141.6004186.02512002144.574183.79282003147.6101181.58732004150.7099179.40832005153.8748177.25542006157.1061175.12832007160.4054173.02682008163.7739170.95052009167.2131168.89912010170.7246166.8723
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