A radioisotope `Y` with half-life `1.4xx10^9` years decays to `X` which is stable. A sample of the rock from a cave was found to contain `X` and `Y` in the ratio `1:7`. The age of the rock is -

(A) 1.96 × 10^9 years.
(B) 3.92 × 10^9 years.
(C) 4.20 × 10^9 years.
(D) 8.40 × 10^9 years.

Options

A. `1.96xx10^9` years
B. `3.92xx10^9` years
C. `4.20xx10^9` years
D. `8.40xx10^9` years

Answer

C

Explanation

Suppose N atoms out of `N_0` atoms of element X disintegrate to element Y in time t.
`therefore N/(N_0-N) =1/7` or , `N/N_0=1/8`
`therefore N/N_0=(1/2)^3`
`therefore` Time taken for this disintegration is 3 times the half-life.
`therefore t=3xx1.4xx10^9 = 4.2xx10^9` years