Download Exponential growth and

Exponential Growth and Decay
MGSE9-12.A.REI.10 & 11
MGSE9-12.F.LE.1-5
Where a = initial amount
b = growth/decay factor
x = time
y = ending amount
Growth
When a > 0 and b > 1,
the function models growth.
(b is called the growth factor)
(a represents the initial amount)
Decay
a > 0 and 0 < b < 1,
When
the function models decay.
(b is called the decay factor)
(a represents the initial amount)
Let's look at examples of these exponential functions at work.
1. Population:
The population of the popular town of Smithville in 2003 was
estimated to be 35,000 people with an annual rate of increase (growth) of about
2.4%.
a.) What is the growth factor for Smithville?
After one year the population would be 35,000 + 0.024(35000).
By factoring we see that this is 35,000(1 + 0.024) or 35,000(1.024).
The growth factor is 1.024. (Remember that the growth factor is greater
than 1.)
b.) Write an equation to model future growth.
where y is the population; x is the number of years
since 2003
c.) Use your equation to estimate the population in 2007 to the nearest hundred
people.
Exponential Growth and Decay
MGSE9-12.A.REI.10 & 11
MGSE9-12.F.LE.1-5
2 Money:
.
Marisa invests $300 at a bank that offers 5% compounded annually.
a.) What is the growth factor for the investment?
The growth factor is 1.05. (Remember that the growth factor is greater
than 1.)
b.) Write an equation to model the growth of the investment.
where y is the money; x is the number of years since the initial
investment
c.) How many years will it take for the initial investment to double?
Algebraic solution:
Graphical solution:
and solve for x.
Answer: approximately 14.2 years
Exponential Growth and Decay
MGSE9-12.A.REI.10 & 11
MGSE9-12.F.LE.1-5
3. Depreciation:
Matt bought a new car at a cost of $25,000. The car
depreciates approximately 15% of its value each year.
a.) What is the decay factor for the value of this car?
The decay factor is 1 - 0.15 = 0.85. (Remember that the decay factor is
0 < b < 1.)
b.) Write an equation to model the decay value of this car.
where y is the value of the car; x is the number of years since new
purchase
c.) What will the car be worth in 10 years?
Answer: $4,921.86
Exponential Growth and Decay
MGSE9-12.A.REI.10 & 11
MGSE9-12.F.LE.1-5
Where y0 = initial amount
k = constant of proportionality
t = time
y = ending amount
Growth
Decay
models growth since e > 1.
Exponential functions with the base e are often
used to describe continuous growth or decay.
models decay since
between 0 and 1
which is
4 More Money: Most banks compound interest more than once a year. When
. interest is compounded n times per year for t years at an interest rate of r, the
principal, P, grows to the amount A given by the formula:
If you allow n to get larger and larger, If compounding takes place
you will discover that as n increases, without interruption (called
compounded continuously),
this formula becomes:
approaches e.
Compound Quarterly:
Jose invests $500 at a bank offering 10%
compounded quarterly. Find the amount of
the investment at the end of 5 years (if
untouched).
Compound Continuously:
Tamika invests $500 at a bank offering 10%
compounded continuously. Find the amount
of the investment at the end of 5 years (if
untouched).
Exponential Growth and Decay
MGSE9-12.A.REI.10 & 11
MGSE9-12.F.LE.1-5
5. Half-Life:
Radium-226, a common isotope of radium, has a half-life of
1620 years. Professor Korbel has a 120 gram sample of radium-226 in his
laboratory.
a.) Find the constant of proportionality for radium-226.
It should make sense that k is a negative
value,
since this is an example of decay.
b.) How many grams of the 120 gram sample will remain after 100 years?
The "official" equation for radium-226 is now
written using the value for k. Substitute the
time and solve.
Exponential Growth and Decay
MGSE9-12.A.REI.10 & 11
MGSE9-12.F.LE.1-5
6. Bacteria Growth:
A certain strain of bacteria that is growing on your
kitchen counter doubles every 5 minutes. Assuming that you start with only one
bacterium, how many bacteria could be present at the end of 96 minutes?
Now, form your equation using this k value,
and solve the problem using the time of 96
minutes.