The half-life of a radioactive substance gives the rate at which the
substance reduces to half of the quantity it is initially.
- The age of the bones at the time they were discovered is approximately 14797 years.
Reasons:
The half life of the radioactive element, [tex]{t_{1/2}}[/tex] = 5750 years
Percentage of the carbon-14 lost = 83.2%
The half-life formula is presented as follows;
[tex]N(t) = N_0 \cdot \left (\dfrac{1}{2} \right )^{\dfrac{t}{t_{1/2}}[/tex]
The percentage of carbon-14 remaining = 100% - 83.2% = 16.8%
Which gives;
[tex]\displaystyle \frac{ N(t) }{N_0} = 0.168 = \left (\dfrac{1}{2} \right )^{\dfrac{t}{5750}[/tex]
[tex]ln( 0.168) = {\dfrac{t}{5750} \cdot ln \left (\dfrac{1}{2} \right )^[/tex]
[tex]t = \displaystyle \frac{ln( 0.168) }{ ln \left (\dfrac{1}{2} \right )} \times 5750 \approx \mathbf{14,797.43}[/tex]
The age of the bones at the time they were discovered, t ≈ 14797.43 years
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