Answer :
Step-by-step explanation:
a.
Initial mass of the isotope = x
Time taken by the sample to decay its mass by 41% = t
Formula used :
[tex]N=N_o\times e^{-\lambda t}\\\\\lambda =\frac{0.693}{t_{\frac{1}{2}}}[/tex]
where,
[tex]N_o[/tex] = initial mass of isotope = x
N = mass of the parent isotope left after the time, (t) = 59% of x = 0.59x
[tex]t_{\frac{1}{2}}[/tex] = half life of the isotope = 4.5 billion years
[tex]\lambda[/tex] = rate constant
Now put all the given values in this formula, we get
[tex]0.59x=x\times e^{-(\frac{0.693}{\text{4.5 billion years}})\times t}[/tex]
t = 3.4 billion years
The age a rock is 3.4 billion years.
b.
Initial mass of the isotope = x
Time taken by the sample to decay its mass by 35%= t
Formula used :
[tex]N=N_o\times e^{-\lambda t}\\\\\lambda =\frac{0.693}{t_{\frac{1}{2}}}[/tex]
where,
[tex]N_o[/tex] = initial mass of isotope = x
N = mass of the parent isotope left after the time, (t) = 65% of x = 0.65x
[tex]t_{\frac{1}{2}}[/tex] = half life of the isotope = 4.5 billion years
[tex]\lambda[/tex] = rate constant
Now put all the given values in this formula, we get
[tex]0.65x=x\times e^{-(\frac{0.693}{\text{4.5 billion years}})\times t}[/tex]
t = 2.8 billion years
The age a rock is 2.8 billion years.