1. LSP 120
Exponential Modeling

2. What Makes It Exponential?
Linear relationship – where a fixed change in x increases or decreases y by a fixed amount
Exponential relationship – for a fixed change in x, there is a fixed percent change in y

3. Exponential, Linear, or Neither?
x y Percent change
no formula here
=(B3-B2)/B2
2 48
3 24
Apply =(B3-B2)/B2. If the column is constant, then the relationship is exponential.
x y Percent change
0 192 % % %
Note: Answer was -.5
but we converted these
cells to % by clicking on
the % icon on the toolbar

4. Linear, Exponential or Neither?
Is each of these linear, exponential, or neither?
x y
5 0.5
10 1.5
15 4.5
x y
0 0
1 1
2 4
3 9
x y
0 0
1 5
2 9
3 13
x y
0 192
1 96
2 48
3 24
Recall: to determine if it is linear: =(B3-B2)/(A3-A2)
To determine if it is exponential: =(B3-B2)/B2

6. General Equation
As with linear, there is a general equation for an exponential function
y = A * (1 + p)x
where A is the initial value of y when x = 0
p is the percent change (written as a decimal)
x is the input variable (very often time)
The equation for the previous example is
y = 192 * (1 + (-0.5))x, or
y = 192 * .5x

7. Exponential Growth
If the percentage change p is greater than 0, then we call the relationship exponential growth
If the percentage change p is less than 0, we call the relationship exponential decay
Many exponentials grow (or decrease) very rapidly eventually, but they also can be very, very flat, sometimes deceptively so.

8. Exponential Growth
If a quantity grows by a fixed percentage change, it grows exponentially
Say a quantity grows by p% each year. After one year, A will become A + Ap, or A*(1+p)
After the second year, you multiply again by 1+p, or A*(1+p)2
After n years, it is A*(1+p)n (look familiar?)

9. Exponential Relationships
For example, the US population is growing by about 0.8% each year. In 2000, the population was 282 million.
A B C
Year Population Population
by adding by multiplying
percent by growth factor (preferred form)
2001 =B2+B2*0.008 =C2*(1+.008)
2002
2003

10. Exponential Relationships
What if a country’s population was decreasing by 0.2% per year?
A B C
Year Population Population
by adding by multiplying
percent by growth factor (preferred form)
2001 =B2-B2*0.002 =C2*(1+(-.002))
2002
2003

11. A Very Common Exponential Growth
Every time you buy something you pay sales tax (let’s say its 8.5%)
The item you purchase is $39.00
Total price = 39 * ( )1
Total price = , or $42.32

12. Another Example
A bacteria population is at 100 and is growing by 5% per minute
How many bacteria cells are present after one hour (60 minutes)?
You could solve it using a spreadsheet…

13. X minutes Y population 100 1 =B2 * (1 + .05) 2 3 =A5+1
100 1
=B2 * ( ) 2 3
=A5+1
Let’s make this chart in Excel.

14. Another Example
Or you could skip the spreadsheet and solve it mathematically
Population = 100 * (1+.05)60
Note: 60 is an exponent in the above equation (but not in the spreadsheet)

15. What If?
What if you knew the final population and wanted to figure out how long it would take to arrive at this answer? For example, when will the Y population be = 1000?
You could look at the spreadsheet, but…
You might have to make a guess
Let’s not guess, let’s use math

16. What If? Let’s use logarithms 1000 = 100(1+.05)X 10 = 1.05X
log(10) = log(1.05)X
log(10) = x * log(1.05)
1 = x * 0.021
x = 1/0.021
x = 47.61

17. Is Exponential Modeling Useful?
Populations tend to grow exponentially
When an object cools, the temperature decreases exponentially towards the ambient temp
Radioactive substances (both good and bad) decay exponentially
Money accumulating in a bank at a fixed rate of interest increases exponentially
Viruses and even rumors spread exponentially

18. Radioisotope Dating What is radioactivity? What is it good for?
What is the connection with exponential growth / decay?

19. Radioisotope Dating So what is radioisotope dating?
The radioisotope age of a specimen is obtained from a calculation of the time that would be required for unstable parent atoms [P] to spontaneously convert to daughter atoms [D] in sufficient amount to account for the present D/P ratio in the specimen.
Unstable Carbon, Uranium, Potassium often used (Carbon 14 (12), Uranium 238, Potassium 40(39))

20. Example
The Dead Sea Scrolls have about 78% of the normally occurring amount of Carbon 14 in them
Carbon 14 decays at a rate of about 1.202% every 100 years
Let’s create a spreadsheet which calculates this exponential delay

21. Example ? Years after Death % Carbon Remaining 0 100
100 =B2 * ( )
200
Stop when % Carbon
Remaining = 78% ?
Let’s go to the lab.