After 4.3 × 106 years, number of produced boron nuclei/ number of remaining beryllium nuclei = 7 . Show that the half-life of beryllium-10 is 1.4 × 106 years

The problem is based on the concept of radioactive decay. The number of remaining nuclei after a certain time is given by the formula:

N = N0 * (1/2)^(t/T)

where: N is the final amount of the substance, N0 is the initial amount, t is the time that has passed, and T is the half-life of the substance.

In this case, we know that after 4.3 × 10^6 years, the ratio of the number of produced boron nuclei to the number of remaining beryllium nuclei is 7. This means that for every 7 boron nuclei, there is 1 beryllium nucleus left.

So, if we started with 8 beryllium nuclei (7 that decayed into boron and 1 that remained), we can set up the equation as follows:

1 = 8 * (1/2)^(4.3 × 10^6 / T)

Solving for T gives:

T = 4.3 × 10^6 / log2(8)

T = 4.3 × 10^6 / 3

T = 1.43 × 10^6 years

So, the half-life of beryllium-10 is approximately 1.4 × 10^6 years.