Respuesta :
Complete Question
Situation: A 25 gram sample of a substance used for drug research has a k-value of 0.1205. [tex]N=N_0e^-kt[/tex]
- [tex]N_0[/tex]= initial mass(at time t=0)
- N= mass at a time t
- K= a positive constant that depends on the substance itself and on the units used to measure time
- T=time, in days
Find the substance's half-life in days, round to the nearest tenth.
Answer:
5.8 days
Step-by-step explanation:
The decay model for drugs and radioactive substances is given as
[tex]N=N_0e^{-Kt}[/tex]
The half-life of any substance is the time it takes for the substance to decay to half its initial amount. That is the period it takes for:
[tex]N(t)=\dfrac{N_0}{2}[/tex]
If we substitute this into the model, we obtain:
[tex]\dfrac{N_0}{2}=N_0e^{-Kt}\\$Dividing both sides by N_o\\\dfrac12=e^{-Kt}[/tex]
We can solve for t.
Taking the natural logarithm of both sides
[tex]\ln\dfrac12=\ln e^{-Kt}\\\implies-\ln2=-Kt\\t=\dfrac{\ln2}{k} \\$Since k=0.1205$\\t_{1/2}=\dfrac{\ln2}{0.1205}=5.8$ days (to the nearest tenth)[/tex]
