Use differential equations to find the amount of radium after 100 years

by RoRi

January 31, 2016

Use an appropriate first-order differential equation to find the percentage of a given quantity of radium which remains after 100 years if we are told that the half-life of radium is 1600 years.

We know (example 1 on page 313 of Apostol) that radioactive decay is modeled by the first-order differential equation

$y' = -ky.$

All solutions to this equation are of the form $f(t) = f(0) e^{-kt}$ . We are given that

$f(1600) = \frac{1}{2}.$

Therefore,

$f(t) = e^{-kt} \quad \implies \quad e^{-1600k} = \frac{1}{2}.$

Hence,

$-k = -\frac{1}{1600} \log 2 \quad \implies \quad k = \log 2^{\frac{1}{1600}}.$

Therefore,

$f(100) = e^{-\log 2^{\frac{1}{16}}} = 2^{-\frac{1}{16}}.$

Thus, the percentage remaining after 100 years is

$100(1-2^{\frac{1}{16}}) = 4.2\%$

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Use differential equations to find the amount of radium after 100 years

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