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 :
![N=N_o\times e^{-\lambda t}\\\\\lambda =\frac{0.693}{t_{\frac{1}{2}}}](https://tex.z-dn.net/?f=N%3DN_o%5Ctimes%20e%5E%7B-%5Clambda%20t%7D%5C%5C%5C%5C%5Clambda%20%3D%5Cfrac%7B0.693%7D%7Bt_%7B%5Cfrac%7B1%7D%7B2%7D%7D%7D)
where,
= initial mass of isotope = x
N = mass of the parent isotope left after the time, (t) = 59% of x = 0.59x
= half life of the isotope = 4.5 billion years
= rate constant
Now put all the given values in this formula, we get
![0.59x=x\times e^{-(\frac{0.693}{\text{4.5 billion years}})\times t}](https://tex.z-dn.net/?f=0.59x%3Dx%5Ctimes%20e%5E%7B-%28%5Cfrac%7B0.693%7D%7B%5Ctext%7B4.5%20billion%20years%7D%7D%29%5Ctimes%20t%7D)
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 :
![N=N_o\times e^{-\lambda t}\\\\\lambda =\frac{0.693}{t_{\frac{1}{2}}}](https://tex.z-dn.net/?f=N%3DN_o%5Ctimes%20e%5E%7B-%5Clambda%20t%7D%5C%5C%5C%5C%5Clambda%20%3D%5Cfrac%7B0.693%7D%7Bt_%7B%5Cfrac%7B1%7D%7B2%7D%7D%7D)
where,
= initial mass of isotope = x
N = mass of the parent isotope left after the time, (t) = 65% of x = 0.65x
= half life of the isotope = 4.5 billion years
= rate constant
Now put all the given values in this formula, we get
![0.65x=x\times e^{-(\frac{0.693}{\text{4.5 billion years}})\times t}](https://tex.z-dn.net/?f=0.65x%3Dx%5Ctimes%20e%5E%7B-%28%5Cfrac%7B0.693%7D%7B%5Ctext%7B4.5%20billion%20years%7D%7D%29%5Ctimes%20t%7D)
t = 2.8 billion years
The age a rock is 2.8 billion years.