Exponential growth and decay

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Transcript of Exponential growth and decay
EXPONENTIAL GROWTH MODEL
WRITING EXPONENTIAL GROWTH MODELS
A quantity is growing exponentially if it increases by the same percent in each time period.
C is the initial amount. t is the time period.
(1 + r) is the growth factor, r is the growth rate.
The percent of increase is 100r.
y = C (1 + r)t
Finding the Balance in an Account
COMPOUND INTEREST You deposit $500 in an account that pays 8% annual interest compounded yearly. What is the account balance after 6 years?
SOLUTION METHOD 1 SOLVE A SIMPLER PROBLEM
Find the account balance A1 after 1 year and multiply by the growth factor to
find the balance for each of the following years. The growth rate is 0.08, so the growth factor is 1 + 0.08 = 1.08.
•••
•••
A1 = 500(1.08) = 540 Balance after one year
A2 = 500(1.08)(1.08) = 583.20 Balance after two years
A3 = 500(1.08)(1.08)(1.08) = 629.856
A6 = 500(1.08) 6 793.437
Balance after three years
Balance after six years
EXPONENTIAL GROWTH MODEL
C is the initial amount.
t is the time period.
(1 + r) is the growth factor, r is the growth rate.
The percent of increase is 100r.
y = C (1 + r)t
EXPONENTIAL GROWTH MODEL
500 is the initial amount. 6 is the time period.
(1 + 0.08) is the growth factor, 0.08 is the growth rate.
A6 = 500(1.08) 6 793.437 Balance after 6 years
A6 = 500 (1 + 0.08) 6
SOLUTION METHOD 2 USE A FORMULA
Finding the Balance in an Account
COMPOUND INTEREST You deposit $500 in an account that pays 8% annual interest compounded yearly. What is the account balance after 6 years?
Use the exponential growth model to find the account balance A. The growthrate is 0.08. The initial value is 500.
Writing an Exponential Growth Model
A population of 20 rabbits is released into a wildlife region. The population triples each year for 5 years.
So, the growth rate r is 2 and the percent of increase each year is 200%.So, the growth rate r is 2 and the percent of increase each year is 200%.So, the growth rate r is 2 and the percent of increase each year is 200%.
1 + r = 31 + r = 3
Writing an Exponential Growth Model
A population of 20 rabbits is released into a wildlife region. The population triples each year for 5 years.
a. What is the percent of increase each year?
SOLUTION
The population triples each year, so the growth factor is 3.
1 + r = 3
The population triples each year, so the growth factor is 3.
Reminder: percent increase is 100r.
A population of 20 rabbits is released into a wildlife region. The population triples each year for 5 years.
b. What is the population after 5 years?
Writing an Exponential Growth Model
SOLUTION
After 5 years, the population is
P = C(1 + r) t Exponential growth model
= 20(1 + 2) 5
= 20 • 3 5
= 4860
Help
Substitute C, r, and t.
Simplify.
Evaluate.
There will be about 4860 rabbits after 5 years.
A Model with a Large Growth Factor
GRAPHING EXPONENTIAL GROWTH MODELS
Graph the growth of the rabbit population.
SOLUTION Make a table of values, plot the points in a coordinate plane, and draw a smooth curve through the points.
t
P 486060 180 540 162020
51 2 3 40
0
1000
2000
3000
4000
5000
6000
1 72 3 4 5 6Time (years)
Po
pu
lati
on
P = 20 ( 3 ) t Here, the large
growth factor of 3 corresponds to a rapid increase
Here, the large growth factor of 3 corresponds to a rapid increase
WRITING EXPONENTIAL DECAY MODELS
A quantity is decreasing exponentially if it decreases by the same percent in each time period.
EXPONENTIAL DECAY MODEL
C is the initial amount.t is the time period.
(1 – r ) is the decay factor, r is the decay rate.
The percent of decrease is 100r.
y = C (1 – r)t
Writing an Exponential Decay Model
COMPOUND INTEREST From 1982 through 1997, the purchasing powerof a dollar decreased by about 3.5% per year. Using 1982 as the base for comparison, what was the purchasing power of a dollar in 1997?
SOLUTION Let y represent the purchasing power and let t = 0 represent the year 1982. The initial amount is $1. Use an exponential decay model.
= (1)(1 – 0.035) t
= 0.965 t
y = C (1 – r) t
y = 0.96515
Exponential decay model
Substitute 1 for C, 0.035 for r.
Simplify.
Because 1997 is 15 years after 1982, substitute 15 for t.
Substitute 15 for t.
The purchasing power of a dollar in 1997 compared to 1982 was $0.59.
0.59
Graphing the Decay of Purchasing Power
GRAPHING EXPONENTIAL DECAY MODELS
Graph the exponential decay model in the previous example. Use the graph to estimate the value of a dollar in ten years.
SOLUTION Make a table of values, plot the points in a coordinate plane, and draw a smooth curve through the points.
0
0.2
0.4
0.6
0.8
1.0
1 123 5 7 9 11Years From Now
Pu
rch
asin
g P
ow
er
(do
lla
rs)
2 4 6 8 10
t
y 0.8370.965 0.931 0.899 0.8671.00
51 2 3 40
0.70.808 0.779 0.752 0.726
106 7 8 9
Your dollar of today will be worth about 70 cents in ten years.
Your dollar of today will be worth about 70 cents in ten years.
y = 0.965t
Help
GRAPHING EXPONENTIAL DECAY MODELS
EXPONENTIAL GROWTH AND DECAY MODELS
y = C (1 – r)ty = C (1 + r)t
EXPONENTIAL GROWTH MODEL EXPONENTIAL DECAY MODEL
1 + r > 1 0 < 1 – r < 1
CONCEPT
SUMMARY
An exponential model y = a • b t represents exponential
growth if b > 1 and exponential decay if 0 < b < 1.C is the initial amount.t is the time period.
(1 – r) is the decay factor, r is the decay rate.
(1 + r) is the growth factor, r is the growth rate. (0, C)(0, C)
Exponential Growth & Decay Models
• A0 is the amount you start with, t is the time, and k=constant of growth (or decay)
• f k>0, the amount is GROWING (getting larger), as in the money in a savings account that is having interest compounded over time
• If k<0, the amount is SHRINKING (getting smaller), as in the amount of radioactive substance remaining after the substance decays over time
ktoA A t A e
Graphs
00
k
eAA kt
A0A0
0
0
ktA A e
k
Example• Population Growth of the United States. In
1990 the population in the United States was about 249 million and the exponential growth rate was 8% per decade. (Source: U.S. Census Bureau)– Find the exponential growth function.– What will the population be in 2020?– After how long will the population be double
what it was in 1990?
Solution• At t = 0 (1990), the population was about 249 million. We
substitute 249 for A0 and 0.08 for k to obtain the exponential growth function.
A(t) = 249e0.08t
• In 2020, 3 decades later, t = 3. To find the population in 2020 we substitute 3 for t:
A(3) = 249e0.08(3) = 249e0.24 317.
The population will be approximately 317 million in 2020.
Solution continued• We are looking for the doubling time T.
498 = 249e0.08T
2 = e0.08T
ln 2 = ln e0.08T
ln 2 = 0.08T (ln ex = x)
= T
8.7 T
The population of the U.S. will double in about 8.7 decades or 87 years. This will be approximately in 2077.
ln 2
0.08
Exponential Decay• Decay, or decline, is represented by the function A(t) =
A0ekt, k < 0.
In this function:
A0 = initial amount of the substance, A = amount of the substance left after time, t = time, k = decay rate.
• The halflife is the amount of time it takes for half of an amount of substance to decay.
Example Carbon Dating. The radioactive element carbon14 has a
halflife of 5715 years. If a piece of charcoal that had lost 7.3% of its original amount of carbon, was discovered from an ancient campsite, how could the age of the charcoal be determined?
Solution: The function for carbon dating is
A(t) = A0e0.00012t.
If the charcoal has lost 7.3% of its carbon14 from its initial amount A0, then 92.7%A0 is the amount present.
Example continuedTo find the age of the charcoal, we solve the equation for t :
The charcoal was about 632 years old.
0.000120 092.7% tA A e
0.000120.927 te0.00012ln 0.927 ln te
ln 0.927 0.00012t
ln 0.927
0.00012632
t
t