Respuesta :
Answer:
It will take 50 years to decay from 512 grams to 121.5 grams.
Step-by-step explanation:
The decay formula :
[tex]N=N_0e^{-\lambda t}[/tex]
where
N= amount of substance after t time
N₀= initial of substance
t= time.
A substance decays at a rate 25% every 10 years.
So, remaining amount of the substance is = (100%-25%)= 75%
[tex]\frac{N}{N_0}=\frac{75\%}{100\%}=\frac{75}{100}=\frac34[/tex], t= 10
[tex]N=N_0e^{-\lambda t}[/tex]
[tex]\Rightarrow \frac {N}{N_0}=e^{-\lambda t}[/tex]
[tex]\Rightarrow \frac34 =e^{-\lambda .10}[/tex]
Taking ln both sides
[tex]\Rightarrow ln|\frac34| =ln|e^{-\lambda .10}|[/tex]
[tex]\Rightarrow ln|\frac34|=-10\lambda[/tex]
[tex]\Rightarrow \lambda=\frac{ ln|\frac34|}{-10}[/tex]
Now , N₀= 512 grams, N= 121.5 grams, t=?
[tex]N=N_0e^{-\lambda t}[/tex]
[tex]\therefore 121.5=512e^{-\frac{ln|\frac34|}{-10}.t}[/tex]
[tex]\Rightarrow 121.5=512e^{\frac{ln|\frac34|}{10}.t}[/tex]
[tex]\Rightarrow \frac{121.5}{512}=e^{\frac{ln|\frac34|}{10}.t}[/tex]
Taking ln both sides
[tex]\Rightarrow ln|\frac{121.5}{512}|=ln|e^{\frac{ln|\frac34|}{10}.t}|[/tex]
[tex]\Rightarrow ln|\frac{121.5}{512}|={\frac{ln|\frac34|}{10}.t}[/tex]
[tex]\Rightarrow t=\frac{ln|\frac{121.5}{512}|}{\frac{ln|\frac34|}{10}}[/tex]
[tex]\Rightarrow t=\frac{10.ln|\frac{121.5}{512}|}{{ln|\frac34|}}[/tex]
⇒t=50 years
It will take 50 years to decay from 512 grams to 121.5 grams.
