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Continuation of exponential modeling and applications. Continuation of exponential modeling and applications.
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Exponential Modeling
The basic function:
How we model real life situations depends on what kind, or how much , information we are given:
Case 1: Working with a minimal amount of information (A,Ao, ∆t).We will create a model in base 10 and base e ... base e is prefered.
is the original amount of "substance" at the beginning of the time period.
A is the amount of "substance" as the end of the time period.
Model is our model for the growth (or decay) of the substance", it is usually an exponential expression in base 10 or base e although any base can be used.
t is the amount of time that has passed for the substance" to grow(or Decay) from to A.
Example: The population of the earth was 5.3 billion in 1990. In 2000 it was 6.1 billion.
(a) Model the population growth using an exponential function.
(b) What was the population in 2008?World Population Clock
(b) What will be the population in 2008?
Case 2: Given lots of information ( , m, p)
A is the amount of "substance " at the end of the time period.
is the original amount of "substance" at the beginning of the time period.
m is the "multiplication factor"or growth rate.
p is the period; the amount of time required to multiply by "m" once.
t is the time that has passed.
Example 1: A colony of bacteria doubles every 6 days. If there were 3000 bacteria to begin with how many bacteria will there be in 15 days?
Example 2: The mass (in grams) of radioactive material in a sample is given by:
where t is measured in years.
(a) Find the half-life of this radioactive substance.
(b) Create a model using the half-life you found in (a). How much of a 10 gram sample of the material will remain after 40 years?
Example 2: The mass (in grams) of radioactive material in a sample is given by:
where t is measured in years.
(b) Create a model using the half-life you found in (a). How much of a 10 gram sample of the material will remain after 40 years?
A $5000 investment earns interest at the annual rate of 8.4% compounded monthly.
a) What is the investment worth after one year?
b) What is it worth after 10 years?
c) How much interest is earned in 10 years?
