Page 30 - Advanced engineering mathematics

10 CHAPTER 1 First-Order Differential Equations

This is an estimate, because an educated guess was made of the body’s temperature before death.
It is also impossible to keep the room at exactly 68 . However, the model is robust in the sense
◦
that small changes in the body’s normal temperature and in the constant temperature of the room
yield small changes in the estimated time of death. This can be verified by trying a slightly
different normal temperature for the body, say 99.3 , to see how much this changes the estimated
◦
time of death.

EXAMPLE 1.6 Radioactive Decay and Carbon Dating
In radioactive decay, mass is lost by its conversion to energy which is radiated away. It has
been observed that at any time t the rate of change of the mass m(t) of a radioactive element
is proportional to the mass itself. This means that, for some constant of proportionality k that is
unique to the element,
dm
= km.
dt
Here k must be negative, because the mass is decreasing with time.
This differential equation for m is separable. Write it as
1
dm = kdt.
m
A routine integration yields
ln|m|= kt + c.

Since mass is positive, |m|= m and
m(t) = e kt+c = Ae kt

in which A can be any positive number. Any radioactive element has its mass decrease according
to a rule of this form, and this reveals an important characteristic of radioactive decay. Suppose
at some time τ there are M grams. Look for h so that, at the later time τ + h, exactly half of this
mass has radiated away. This would mean that

M
kτ kh
m(τ + h) = = Ae k(τ+h) = Ae e .
2
kτ
But Ae = M, so the last equation becomes
M
kh
= Me .
2
Then
1
kh
e = .
2
Take the logarithm of this equation to solve for h, obtaining
1
1 1
h = ln =− ln(2).
k 2 k
This is positive because k < 0.
Notice that h, the time it takes for half of the mass to convert to energy, depends only
on the number k, and not on the mass itself or the time at which we started measuring the
loss. If we measure the mass of a radioactive element at any time (say in years), then h years
later exactly half of this mass will have radiated away. This number h is called the half-life

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October 14, 2010 14:9 THM/NEIL Page-10 27410_01_ch01_p01-42