Math 151
Section 4.5
Exponential Growth and Decay
Exponential Growth and Decay A function grows or decays exponentially when the rate of
change is proportional to the original function for some constant k, i.e.,
dy
= ky
dt
In particular, if k > 0, the function grows and if k < 0, the function decays.
Verify that the solution to the differential equation is y (t ) = Aekt where y (0) = A .
Note: The constant of proportionality k is the coefficient of t in the exponent and A is the initial
value of y.
Example: A bacteria culture starts with 1000 bacteria and 3 hours later it has 3000 bacteria.
Assuming that the growth of the culture is proportional to its size.
A. Find a function to model the population growth.
B. How many bacteria are present after 9 hours?
C. How long does it take the population to double?
Math 151
Half-Life The half-life of a radioactive substance is the amount of time it takes for half of the
substance to decay. The rate of decay is proportional to the amount of substance present.
Example: Strontium-90 has a half-life of about 29 years.
A. Find a function that could be used to determine the amount of strontium-90 remaining at time t
for a given sample.
B. How much will be left after 14 years?
C. How long until only one-eighth of the sample remains?
D. How long until 60% of the sample is gone?
Example: Suppose that a radioactive substance decays to 48% of its original amount in 12 days.
What is the half-life of the substance?
Math 151
Newton’s Law of Cooling Newton’s law of cooling states that the rate of cooling of an object is
proportional to the temperature difference between the object and its surroundings. Let y represent
the temperature of the object and let A represent the temperature of the room. Then y ! = k ( y " A) .
Example: A potato is taken from the oven when it has reached 190°F and is placed in a room with a
temperature of 70°F. If the temp of the potato is 160°F after half an hour, what is its temperature
after 45min?