A radio isotope 'X' with a half life 1.4 × 10⁹ years decays to 'Y' 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

Let N₀ be the initial amount of X. After time t, the amount of X remaining is given by:
N = N₀(1/2)^(t/T), where T is the half-life.
The ratio of X to Y is 1:7. This means that for every 1 part X remaining, there are 7 parts Y. Since Y is formed from the decay of X, the initial amount of X was 1 + 7 = 8 parts.
Therefore, the fraction of X remaining is 1/8.
We can set up the equation:
1/8 = (1/2)^(t/T)
Taking the logarithm of both sides (base 1/2):
log(1/2)(1/8) = t/T
3 = t/T
t = 3T
Given that the half-life T = 1.4 × 10⁹ years,
t = 3 × 1.4 × 10⁹ years = 4.2 × 10⁹ years
Therefore, the age of the rock is 4.2 × 10⁹ years.