Respuesta :
Answer:
a) The half life of the element is 231 days.
b) It is going to take around 31.5 days for a sample of 100 mg to decay to 91 mg.
Step-by-step explanation:
The radioactivity of the sample can be modeled by the following exponential equation:
[tex]R(t) = R(0)e^{-rt}[/tex]
In which t is the time in days, r is the decay rate and [tex]R(0)[/tex] is the initial radioactive percentage.
We have that:
In 440 days the radioactivity of a sample decreases by 74 percent.
This means that [tex]R(440) = 0.26R(0)[/tex].
This helps us find r.
[tex]R(t) = R(0)e^{-rt}[/tex]
[tex]0.26R(0) = R(0)e^{-440r}[/tex]
[tex]e^{-440r} = 0.26[/tex]
Applying ln to both sides.
[tex]\ln{e^{-440r}} = \ln{0.26}[/tex]
[tex]-440r = -1.347[/tex]
[tex]r = 0.003[/tex]
(a) What is the half-life of the element?
This is t when [tex]R(t) = 0.50R(0)[/tex]
[tex]R(t) = R(0)e^{-rt}[/tex]
[tex]0.50R(0) = R(0)e^{-0.003t}[/tex]
[tex]e^{-0.003t} = 0.5[/tex]
Again, we apply ln to both sides of the equality.
[tex]\ln{e^{-0.003t}} = \ln{0.5}[/tex]
[tex]-0.003t = -0.693[/tex]
[tex]t = 231[/tex]
The half life of the element is 231 days.
(b) How long will it take for a sample of 100 mg to decay to 91 mg?
This is t when [tex]R(t) = 0.91R(0)[/tex]
[tex]R(t) = R(0)e^{-rt}[/tex]
[tex]0.91R(0) = R(0)e^{-0.003t}[/tex]
[tex]e^{-0.003t} = 0.91[/tex]
[tex]\ln{e^{-0.003t}} = \ln{0.91}[/tex]
[tex]-0.003t = -0.09[/tex]
[tex]t = 31.44[/tex]
It is going to take around 31.5 days for a sample of 100 mg to decay to 91 mg.
