7.5 Exponential Growth and Decay
Quantities
grow or decay at a rate proportional to their size
If y is exponential function then 
would satisfy the
differential equation, since
Theorem
The only
solutions of the differential equation are the exponential
functions
Example –
Exponential Growth
#6 Table on
pg 453 gives the population for the US in millions for 1900-2000
a) Use the exponential model and the census
figures for 1900 (76) and 1910 (92) to
predict the population in 2000. Compare
with the actual figure (275).
b) Use the exponential model and the census
figures for 1980 (227) and 1990 (250) to predict the population for 2000. Compare the actual population (275).
Example
– Exponential Decay
#8 Bismuth-210 has a half-life of 5.0 days.
a) A sample originally has a mass of 800
mg. Find a formula for the mass
remaining after t day.
b) Find the mass remaining after 30 days.
c) When is the mass reduced to 1 mg?
#10 A sample of tritium-3
decayed to 94.5% of its original amount after a year.
a) What is the half-life of tritium-3?
b) How long would it take the sample to decay to
20% of its original amount?
Newton’s law of Cooling
This states
that the rate of cooling of an object is proportional to the temperature
difference between the object and its surroundings, provided that the
difference is not to large. Let T(t) be temperature at time t and 
be the temperature of
the surroundings, then
If we let 
Demonstrate
with an example:
#14 A thermometer is
taken from a room where the temperature is 20 degrees C to the out doors, where
the temperature is 5 degrees C. After
one minute the thermometer reads 12 decrees C.
a) What will the reading on the thermometer be
after one more minute?
b) When will its temperature be 15 degrees C.
So 
. Let y = T(t) – 5. and so 
(Note y(0) is difference of initial temp and outside temp)
Use
temperature is 12 when time is 12 to find k.
Then since y = T(t) – 5, find T(t)
= y + 5.
a) What will the reading on the thermometer be after
one more minute?
b) When will its temperature be 15 degrees
C.
Assignment: 7.5
page 453; 1-13 odd
Chapter
Review – page 483; 21, 23, 29, 3139, 57,
93, 97, 99