1.
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.
2.
Example: The population of the earth was 5 billion in 1990. In 200l it was
6.2 billion.
(a) Model the population growth using an exponential function.
3.
Example: The population of the earth was 5 billion in 1990. In 200l it was
6.2 billion.
(a) Model the population growth using an exponential function.
4.
Example: The population of the earth was 5 billion in 1990. In 200l it was
6.2 billion.
(a) Model the population growth using an exponential function.
5.
Example: The population of the earth was 5 billion in 1990. In 200l it was
6.2 billion.
(a) Model the population growth using an exponential function.
(b) What was the population in 1999?
6.
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.
7.
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?
8.
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.
9.
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?
10.
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?
11.
An amount of $3,000.00 is deposited in a bank paying an annual interest rate of
3 %, compounded continuously.
(a) Find the balance after 4 years.
(b) How long would it take for the money to double?
12.
At the present time, there are 1000 type A bacteria. If the rate of increase
per hour is 0.025, how many bacteria can you expect in 24 hours?
13.
A radioactive substance decays at a daily rate of 0.13. How long does it take for
this substance to decompose to half its size?
14.
In 1999, the population of Winnipeg was 630 700, and was increasing at the rate
of 0.54% per year.
(a) Write an equation to represent the population of Winnipeg, P, as a
function of the number of years, y, since 1999.
(b) Calculate how many years it would take for the population to double.
(c) Calculate when the population will reach 1 million.
(d) Write the equation in part (a) as an exponential function with base 2.
(e) Write the equation in part (a) as an exponential function with base e.
15.
In 1999, the population of Winnipeg was 630 700, and was increasing at the rate
of 0.54% per year.
(a) Write an equation to represent the population of Winnipeg, P, as a
function of the number of years, y, since 1999.
16.
In 1999, the population of Winnipeg was 630 700, and was increasing at the rate
of 0.54% per year.
(b) Calculate how many years it would take for the population to double.
17.
In 1999, the population of Winnipeg was 630 700, and was increasing at the rate
of 0.54% per year.
(c) Calculate when the population will reach 1 million.
18.
In 1999, the population of Winnipeg was 630 700, and was increasing at the rate
of 0.54% per year.
(d) Write the equation in part (a) as an exponential function with base 2.
(e) Write the equation in part (a) as an exponential function with base e.
Be the first to comment