Answer :
Answer:
Therefore 10.75 days is the half life of the given radioactive substance.
Step-by-step explanation:
Given that, a radioactive substance decays function is
[tex]y=y_0 \ e^{-0.0645t}[/tex]
where y₀= initial amount of radioactive substance
y= The amount of radioactive substance at time t.
t is in days
Half life is the time required to reduce the amount of the substance to its half of the initial amount.
For half life
[tex]y=\frac{1}{2} y_0[/tex]
Putting the value of y in the given function
[tex]\frac12y_0=y_0 \ e^{-0.0645t}[/tex]
[tex]\Rightarrow \frac12= \ e^{-0.0645t}[/tex]
Taking ln both sides
[tex]\Rightarrow ln( \frac12)= ln(e^{-0.0645t})[/tex]
[tex]\Rightarrow ln1-ln2=-0.0645t[/tex] [ [tex]\because lne^a=a,[/tex][tex]ln(\frac ab)= ln \ a- ln \ b[/tex] ]
[tex]\Rightarrow-ln2=-0.0645t[/tex]
[tex]\Rightarrow t=\frac{-ln2}{-0.0645}[/tex]
=10.75 days.
Therefore 10.75 days is the half life of the given radioactive substance.
The half-life of the substance is approximately 10.7 days.
The decay of a radioactive isotope is represented by the following exponential model:
[tex]y = y_{o}\cdot e^{-\frac{t}{\tau} }[/tex] (1)
Where:
- [tex]y_{o}[/tex] - Initial amount.
- [tex]y[/tex] - Current amount.
- [tex]t[/tex] - Time, in days.
- [tex]\tau[/tex] - Time constant, in days.
The half-life of the substance ([tex]t_{1/2}[/tex]) can be found by this expression:
[tex]t_{1/2} = \tau \cdot \ln 2[/tex] (2)
If we know that [tex]\tau = 15.504[/tex], then the half-life of this substance is:
[tex]t_{1/2} = 15.504\cdot \ln 2[/tex]
[tex]t_{1/2} \approx 10.7[/tex]
The half-life of the substance is approximately 10.7 days.
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