Respuesta :
Answer:
The half-life of this isotope is of 554 years.
Step-by-step explanation:
The amount of the radioactive isotope remaining after t years is given by the following equation:
[tex]P(t) = P(0)(1-r)^{t}[/tex]
In which P(0) is the initial amount and r is the yearly rate that it decays, as a decimal.
A certain radioactive isotope decays at a rate of 0.125% per year.
This means that [tex]r = 0.00125[/tex]
Then
[tex]P(t) = P(0)(1-0.00125)^{t}[/tex]
[tex]P(t) = P(0)(0.99875)^{t}[/tex]
Determine the half-life of this isotope, to the nearest year.
This t for which [tex]P(t) = 0.5P(0)[/tex]
Then
[tex]P(t) = P(0)(0.99875)^{t}[/tex]
[tex]0.5P(0) = P(0)(0.99875)^{t}[/tex]
[tex](0.99875)^{t} = 0.5[/tex]
[tex]\log{(0.99875)^{t}} = \log{0.5}[/tex]
[tex]t\log{0.99875} = \log{0.5}[/tex]
[tex]t = \frac{\log{0.5}}{\log{0.99875}}[/tex]
[tex]t = 554.17[/tex]
Rounding to the neearest year
The half-life of this isotope is of 554 years.
