Radioactive Decay

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 No.  

        t = 0      N = No

        And from the definition of half-life:

        t = 5600      N = No/2

        Also:      dN/dt = kN

    And what is it that we are looking for?

        t = ?      when N = 0.1No

Now, separate variables and integrate.

   dN/N = k dt

    ln N = kt + C

    N = C ekt

Solve for the constants:

Use     t = 0      N = No

    No = Cek(0)

    C = No

    Now use     t = 5600     N = No/2

    No/2 = Noek(5600)

    k = -0.0001238

    So the equation is:         N = Noe-.0001238t

We can now solve for time:

Use  t = ?      when N = 0.1No

    0.1No = Noe-.0001238t

    t = 18,600 years

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