AP AB Calculus:
Half-Lives
Objective
To derive the half-life equation using calculus
To learn how to solve half-life problems
To solve basic and challenging half-life problems
To understand the applications of half-life problems in
real-life
Do Now:
Exponential Growth
Problem:
In 1985, there were 285 cell phone subscribers in the
town of Centerville. The number of subscribers
increased by 75% per year after 1985. How many cell
phone subscribers were in Centerville in 1994?
Answer:
y= a (1 + r ) ^x
y= 285 (1 + .75) ^9
y= 43871 subscribers in 1994
What is a half-life?
The time required for half of a given substance to decay
Time varies from a few
microseconds to billions
of years, depending on
the stability of the
substance
Half-lives can
increase or remain
constant over time
Calculus Concepts
Growth & Decay Derivation
The rate of change of a variable y at time t is proportional to
the value of the variable y at time t, where k is the constant of
proportionality.
dy
dt
ò
= ky
dy
=
y
ò k dt
ln y = kt + C
y =e
kt+C
y = ekt ec
y  Ce
kt
Calculus Concepts Cont.
Therefore, the equation for the amount of a radioactive
element left after time t and a positive k constant is:
y  Ce
kt
The half-life of a substance is found by setting this
equation equal to double the amount of substance.
Calculus Concepts Cont.
Half-life Derivation
2 C  Ce
kt
ln 2  ln( e )
kt
ln 2  kt
ln 2
k
t
Half-life Equation (used
primarily in chemistry):
half-life 
ln 2
k
How to solve a half-life problem
Steps to solve for amount of time t
Use given information to solve for k
Given information: initial amount of substance (C),
half of the final amount of substance (y), half-life of
substance (t)
Use k in the original equation to determine t
Original equation: initial amount of substance (C),
final amount of substance (y), constant of
proportionality (k)
How to solve a half-life problem
Steps to solve for final amount of substance y
Use given information to solve for k
Given information: initial amount of substance
(C), half of the final amount of substance (y),
half-life of substance (t)
Use k in the original equation to determine y
Original equation: initial amount of substance
(C), time elapsed (t), constant of proportionality
(k)
Basic Example #1
Problem: Suppose 10g of plutonium Pu-239 was released
in the Chernobyl nuclear accident. How long will it take
the 10g to decay to 1g? (Half life Pu-239 is 24,360 years.)
Answer:
y  Ce
ln . 5 t
kt
1
(10) =10e24,360k
2
5 =10e24,360k
ln . 5  ln e
k 
1  10 e 24 , 360
ln . 5 t
ln . 1  ln e 24 , 360
ln . 1 
24 ,360
24 , 360 k
ln . 5
24 ,360
ln . 5 t
t
24 ,360 ln . 1
ln . 5
t  80 ,922 . 17 years
Basic Example #2
Problem: Cobalt-60 is a radioactive element used as a
source of radiation in the treatment of cancer. Cobalt-60
has a half-life of five years. If a hospital starts with a 1000mg supply, how much will remain after 10 years?
Answer:
10 ln . 5
y  Ce
1
2
kt
(1000 )  1000 e
.5  e
5
y  1000 e
2 ln . 5
5k
ln . 5  5 k
k 
5k
y  1000 e
ln . 5
5
y  250 mg
Challenging Example #1
Problem: The half-life of Rossidium-312 is 4,801 years. How
long will it take for a mass of Rossidium-312 to decay to
98% of its original size?
Answer:
ln . 5
y  Ce
1
kt
(1)  (1) e
4801 k
. 98  (1) e 4801
ln . 98 
k 
4801 k
ln . 5
4801
t
4801
2
.5  e
ln . 5
t
4801 ln . 98
ln . 5
t  139 . 93 years
Challenging Example #2
Problem: The half-life of carbon-14 is 5730 years. A bone is
discovered which has 30 percent of the carbon-14 found in
the bones of other living animals. How old is the bone?
Answer:
ln . 5 t
y  Ce
1
2
(. 3 )  . 3 e
.5  e
k 
kt
5730 k
5730 k
ln . 5
5730
. 3  (1) e 5730
ln . 3 
ln . 5 t
5730
t  9952 . 81 years
Applications in Real Life
Radioactive decay: half the amount of
time for atoms to decay and form a more
stable element
Knowing the half-life enables one to
date a partially decayed sample
Examples: fossils, meteorites,
carbon-14 in once-living bone and
wood
Biology: half the amount of time
elements are metabolized or eliminated
by the body
Knowing the half-life enables one to
determine appropriate drug dosage
amounts and intervals
Examples: Pharmaceutics, toxins
Summary of Half-Lives
Definition: Time required for something to fall to
half it’s initial value
Calculus Concept: A particular form of exponential
decay
Solve Problems: First solve for
constant of proportionality (k),
then determine unknown variable
Processes of half-lives:
radioactive decay,
pharmaceutical science