Differential Equation Applications

Radioactive Decay

A piece of human bone is found at an archeological site. If 10% of the original amount of radioactive carbon-14 was present, estimate the age of the bone. The half-life of C-14 is 5600 years. (Half-life is the the time it takes for half of it to decay.)

Write down what is known.

We don’t know what the original amount is, so we’ll call it N_{o}.

t = 0 N = N_{o}

And from the definition of half-life:

t = 5600 N = N_{o}/2

Also: dN/dt = kN

And what is it that we are looking for?

t = ? when N = 0.1N_{o}

Now, separate variables and integrate.

dN/N = k dt

ln N = kt + C

N = C e^{kt}

Solve for the constants:

Use t = 0 N = N_{o}

N_{o} = Ce^{k(0)}

C = N_{o}

Now use t = 5600 N = N_{o}/2

N_{o}/2 = N_{o}e^{k(5600)}

^{ }k = -0.0001238

So the equation is: N = N_{o}e^{-.0001238t}

We can now solve for time:

Use t = ? when N = 0.1N_{o}

_{ }0.1N_{o} = N_{o}e^{-.0001238t}

t = 18,600 years