The answer is 6396 years
Let's first calculate a number of half-lives using the equation:
[tex] (1/2)^{n} [/tex]
= decimal amount remaining, where n i number of half-lives
After 15/16 of a given amount of radium-226 has been decayed, that means that the remaining amount is 1/16, which is in decimals 0.0625
So, now we can replace it:
tex] (1/2)^{n} = 0.0625 [/tex]
⇒ n * log(1/2) = log(0.0625)
n * log(0.5) = log(0.0625)
n = log(0.0625)/log(0.5)
n = 4
Now we know that number of half-lives is 4.
The number of half-lives is a quotient of total time elapsed and length of half-life.
So, total time elapsed is a product of a length of half-life (1599 years) and the number of half-lives (4). Since 1599 × 4 = 6396, then a scientist will have to wait 6396 years for decay of 15/16 of a given amount of radium-226.
