The answer is C) 11400 years
Let's first calculate the remaining amount in percent:
212g is original amount or 100%.
53 g is x percents.
212 g : 100% = 53 g : x
x = 100% * 53 g : 212 g
x = 25% = 0.25
Now, using the formula to calculate the number of half-lives:
(1/2)ⁿ = x,
where
x is the remaining amount: x = 0.25
n is the number of half-lives
1/2 stands for half-life.
(1/2)ⁿ = 0.25
⇒ n*log(1/2) = log(0.25)
n = log(0.25)/log(1/2) = log(0.25)/log(0.5) = -0.602/-0.301 = 2
The number of half-lives is 2.
Now, the number of half-lives (n) is a quotient of total time elapsed (T) and length of half-life (L):
n = T/L
We know:
n = 2
L = 5700 years
T = ?
Thus
T = L * n
L = 5700 years * 2
L = 11400 years
