Transcript Exponential Growth and Decay

Review
Write and solve the differential equation that
models the verbal statement. Evaluate the solution
at the specified value.
The rate of change of B is proportional to B. When
B = 2, t = 0 and when B = 8, t = 3. What is the
value of t when B = 10?
3ln 5
t
 3.483
ln 4
Find the exponential function y = y0ek t whose graph passes
through the two points (0, 3) and (2, 10)
Find the exponential function y = y0ek t whose graph passes
through the two points (1, 1.5) and (3, 6)
Exponential Growth or Decay
Any situation where the rate of change of
a function is proportional to the value of
the function…
For example, any kind of population
growth, radioactive decay, etc.
the rate of change of a function, P, is proportional to
the value of the function, P,…in other words
dP
 kP
dt
Use separation of variables to solve…
In 1980 the population of a town was 21,000 and
in 1990 was 20,000. If the population is following
an exponential pattern of decline, what is the
expected population in 2100?
(Let t = 0 be 1980)
Example
Suppose a colony of bacteria is growing (exponentially) in
the corner of your shower stall. On June 1, there are 1
million bacteria there. By July 1, there are 7.5 million. Your
shower stall can contain a total of 1 billion bacteria. When
will you have to start takings showers down at the gym?
The half life of radioactive carbon-14 is 5,750 years.
• There are 10 grams present initially. Write an equation in the
form y = Cekt to represent the amount present at anytime t.
• How much will remain of the 10 grams after 3000 years?
• How long will it take for the 10 grams to be reduced to 1 gram?
The half life of radioactive carbon-14 is 1,250 years.
There are 10 grams present after 500 years. What was the
initial quantity of the substance?
How much will remain after 3000 years?
Hiram Fentley, heir to the Fentley Feta Cheese fortune was found
dead at his home at 2:30 am. The temperature in the room was a
constant 70 degrees. His body temperature at 3:00am was 85 degrees
and at 4:00am was 78 degrees. At what time was he killed?
Newton’s Law of Cooling states that the rate of change in the
temperature of an object is proportional to the difference between the
object’s temperature and the temperature of the surrounding medium.
dT
 k (T  R )
dt
Which when solved is…
T= body temperature
R = temp of the surroundings
t = time
T (t )  Ce  R
kt
Let P(t) represent the number of wolves in a population at time t
years, when t > 0. The population P(t) is increasing at a rate
directly proportional to 800 – P(t), where the constant of
proportionality is k.
1) Write a differential equation that models this
growth.
2)
If P(2) = 700, write an equation P that models
the given situation.
3)
P (t )
Find lim
t 
HOMEWORK CHANGE
worksheet