Respuesta :
Answer:
Time to decay will be 377.7 days.
Step-by-step explanation:
Decay of an radioactive element is represented by the formula
[tex]A_{t}=A_{0}e^{-kt}[/tex]
where [tex]A_{t}[/tex] = Amount after t days
[tex]A_{0}[/tex] = Initial amount
t = duration for the decay
k = decay constant
Now we plug in the values in the formula
[tex](1-0.12)x=xe^{-30k}[/tex]
[tex](0.88)x=xe^{-30k}[/tex]
[tex]0.88=e^{-30k}[/tex]
Now we take natural log on both the sides
ln(0.88) = [tex]ln(e)^{-30k}[/tex]
ln(0.88) = -30k(lne)
-30k = -0.1278
k = [tex]\frac{.1278}{30}[/tex]
k = [tex]4.261\times 10^{-3}[/tex]
Now we have to calculate the duration for the decay of 50 mg sample to 10 mg.
[tex]A_{t}=A_{0}e^{-kt}[/tex]
We plug in the values in the formula
10 = 50[tex]e^{-4.261\times 10^{-3}\times t}[/tex]
[tex]e^{-4.261\times 10^{-3}\times t}=\frac{10}{50}[/tex]
[tex]e^{-4.261\times 10^{-3}\times t}=0.2[/tex]
We take (ln) on both the sides
[tex]ln(e^{-4.261\times 10^{-3}\times t})=ln(0.2)[/tex]
[tex]-4.261\times 10^{-3}\times t=-1.6094[/tex]
t = [tex]\frac{1.6094}{4.261\times 10^{-3} }[/tex]
t = 0.37771×10³
t = 377.7 days
Therefore, time for decay will be 377.7 days.
