Transcript Exponential Modeling

LSP 120
Exponential Modeling
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

Exponential, Linear, or Neither?
x
0
1
2
3
y
192
96
48
24
Percent change
no formula here
=(B3-B2)/B2
Apply =(B3-B2)/B2. If the column is constant, then the relationship is
exponential.
x
0
1
2
3
y
192
96
48
24
Percent change
-50%
-50%
-50%
Note: Answer was -.5
but we converted these
cells to % by clicking on
the % icon on the toolbar
Linear, Exponential or Neither?
Is each of these linear, exponential, or neither?
x
5
10
15
20
y
0.5
1.5
4.5
13.5
x
0
1
2
3
y
0
1
4
9
x
0
1
2
3
y
0
5
9
13
Recall: to determine if it is linear: =(B3-B2)/(A3-A2)
To determine if it is exponential: =(B3-B2)/B2
x
0
1
2
3
y
192
96
48
24
X
Y
Linear?
Exp?
5
10
15
20
0.5
1.5
4.5
13.5
0.2
0.6
1.8
200%
200%
200%
0
1
2
3
0
1
4
9
1
3
5
#DIV/0!
300%
125%
0
1
2
3
0
5
9
13
5
4
4
#DIV/0!
80%
44%
0
1
2
3
192
96
48
24
-96
-48
-24
-50%
-50%
-50%
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
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.

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?)

Exponential Relationships

For example, the US population is growing
by about 0.8% each year. In 2000, the
population was 282 million.
A
Year
B
Population
by adding
percent
C
Population
by multiplying
by growth factor (preferred form)
2000
2001
2002
2003
282
=B2+B2*0.008
282
=C2*(1+.008)
Exponential Relationships

What if a country’s population was
decreasing by 0.2% per year?
A
Year
B
Population
by adding
percent
C
Population
by multiplying
by growth factor (preferred form)
2000
2001
2002
2003
344
=B2-B2*0.002
344
=C2*(1+(-.002))
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+0.085)1
 Total price = 42.315, or $42.32

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…

X minutes
Y population
0
100
1
=B2 * (1 + .05)
2
3
=A5+1
Let’s make this chart in Excel.
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)
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

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
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

Radioisotope Dating
What is radioactivity?
 What is it good for?
 What is the connection with exponential
growth / decay?

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))
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

Example
Years after Death
0
100
200
?
Let’s go to the lab.
% Carbon Remaining
100
=B2 * (1 - 0.01202)
Stop when % Carbon
Remaining = 78%