The half life of Pb-210 is 22 years. A decayed animal shows 16% of the original Pb-210 remains; how long has the animal been deceased to the nearest tenth of a year?

The answer is 8.5 years

Let's use the formula to calculate the number of half-lives:

(1/2) ⁿ = x,

where

x is the remaining amount in decimals: x = 16% = 0.16

n is the number of half-lives

1/2 stands for half-life.

(1/2) ⁿ = 0.16

⇒ n*log (1/2) = log (0.16)

n = log (0.16) / log (1/2) = log (0.16) / log (0.5) = - 0.796/-0.301 ≈ 2.6

The number of half-lives is 2.6.

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 = 2.6

T = 22 years

L = ?

Thus

L = T/n

L = 22 years/2.6 = 8.5 years