Download Lesson 71 Template.notebook

Lesson 71 Template.notebook
March 06, 2012
Growth and Decay Problems
Using Exponential and Logarithmic Functions
• Exponential functions are of the form y = a(bx)
Observe how the graphs of exponential functions change based upon the values of
a and b:
Growth: Example:
when a > 0 and the b is greater than 1,
the graph will be increasing (growing).
For this example, each time x is increased
by 1, y increases by a factor of 2.
Such a situation is called
Exponential Growth.
Decay:
when
a > 0 and the b is between 0 and 1,
the graph will be decreasing (decaying).
For this example, each time x is increased by
1, y decreases to one half of its previous
value.
Such a situation is called
Exponential Decay.
Many real world phenomena can be modeled by functions that describe how
things grow or decay as time passes. Examples of such phenomena include the
studies of populations, bacteria, the AIDS virus, radioactive substances,
electricity, temperatures and credit payments, to mention a few.
Any quantity that grows or decays by a fixed percent at regular intervals is
said to possess exponential growth or exponential decay.
At this level, there are two functions that can be easily used to illustrate the
concepts of growth or decay in applied situations. When a quantity grows by a
fixed percent at regular intervals, the pattern can be represented by the
functions,
Decay:
Growth: a = initial amount before measuring growth/decay
r = growth/decay rate (often a percent, but written as a decimal)
n = number of time intervals that have passed
Example:
A bank account balance, A, for an account starting with P dollars,
earning an annual interest rate, r, and left untouched for t years can be calculated as
(an exponential growth formula). Find a bank account balance to the nearest dollar, if the
account starts with $100, has an annual rate of 4%, and the money left in the
account for 12 years.
The population of Boomville starts off at 20,000, and grows by 13% each
year. Write an exponential growth model and find the population after 10
years.
Kelly plans to put her graduation money into an account and leave it there
for 4 years while she goes to college. She receives $2500 in graduation money
that she puts it into a CD (certificate of deposit) that earns 2.8% interest
compounded semi-annually. How much will be in Kelly’s account at the end
of four years? Mar 5­11:14 AM
Mar 6­8:57 AM
On the last day of this month you find a penny. You
bring it to school on the first day of next month and I
double it for you. Every day after that I give you double
the amount that I gave you the previous day. I continue
this until the end of the month.
Which would you rather have:
A. The amount of money described in this scenario or
B. $5,000,000 ?
Mar 6­8:50 AM
Mar 6­8:40 AM
Decay
Each year the local country club sponsors a
tennis tournament. Play starts with 128
participants. During each round, half of the
players are eliminated. How many players
remain after 5 rounds?
Round
0
Players 128
1
2
3
4
5
64
32
16
8
4
Notice the shape of this graph compared to the graphs of the
growth functions.
Suppose Mark has $1000 that he invests in an account that pays 3.5%
interest compounded quarterly. How much money does Mark have at the
horizontal axis = rounds
vertical axis = number of players left
end of 5 years?
A car bought for $43,000 depreciates at 12% per annum. What is
its value after 7 years?
If a person takes A milligrams of a drug at time 0, then y = A(0.7) t
gives the concentration left in the bloodstream after t hours. If the
initial dose is 125 mg, what is the concentration of the drug in the
bloodstream after 3 hours?
Mrs. Potter wants to have a total of $12,000 in two years so that she can
put a hot tub on her deck. She finds an account that pays 3% interest
compounded monthly. How much should Mrs. Potter put into this account so
that she'll have $12,000 at the end of two years?
Does the equation y = 11(1.11) t model exponential growth or
exponential decay?
Find the growth or decay factor and the percent change per
time period.
Does the equation y = 27(3/2) t model exponential growth or
exponential decay?
Find the growth or decay factor and the percent change per
time period.
Does the equation y = 7(3/4) t model exponential growth or exponential
Suppose Mrs. Potter, from our last example, only has $10,000 to invest but
decay?
still wants $12,000 for a hot tub. She finds a bank offering 3.25% interest
Find the growth or decay factor and the percent change per time
compounded quarterly. How long will she have to leave her money in the
period.
account to have $12,000?
Mar 5­8:39 PM
Mar 5­2:55 PM
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