Answer:
Option A.
Step-by-step explanation:
Half life of Uranium-235 has been given as 700 million years.
Since radioactive decay is an exponential phenomenon so the formula will be
[tex]A_{t}=A_{0}e^{-kt}[/tex]
where [tex]A_{t}[/tex] = Amount of the radioactive element at the time 't'
[tex]A_{0}[/tex] = Initial amount
t = time for decay
k = decay constant
By this formula,
[tex]\frac{A_{0}}{2}=A_{0}e^{-kt}[/tex]
[tex]\frac{1}{2}=e^{-700\times 10^{6} k}[/tex]
By taking natural log on both the sides,
[tex]ln(\frac{1}{2})=ln(e^{-700\times 10^{6}k } )[/tex]
[tex]-ln2=-700\times 10^{6}\times k[/tex]
0.693147 = [tex]700\times 10^{6}k[/tex]
k = [tex]\frac{0.693147}{700\times 10^{6}}[/tex]
= [tex]\frac{0.693147}{7\times 10^{8}}[/tex]
= [tex]9.9\times 10^{-10}[/tex]
Now we have to find the age of the rock which is one sixteenth of the original rock.
By the formula again,
[tex]A_{t}=A_{0}e^{-kt}[/tex]
[tex]\frac{A_{0}}{16}=A_{0}e^{-kt}[/tex]
[tex]\frac{1}{16}=e^{-9.9\times 10^{-10}t}[/tex]
Taking log on both the sides.
[tex]ln\frac{1}{16}=ln(e^{-9.9\times 10^{-10}t})[/tex]
[tex]2.772588=-9.9\times 10^{-10}\times t[/tex]
t = [tex]\frac{2.772588}{9.9\times 10^{-10} }[/tex]
t = [tex]0.28\times 10^{10}[/tex]
t = [tex]2.8\times 10^{9}[/tex]
Therefore, the rock is 2.8 billion years old.
Option A. is the answer.
