Question

The half life of Pb-210 is 22 years. A decayed animal shows 25% of the original Pb-210 remains; how long has the animal been deceased to the nearest tenth of a year?

Answer 1

Answer:

44 years.

Step-by-step explanation:

Let original amount be 100.

We have been given that the half life of Pb-210 is 22 years. A decayed animal shows 25% of the original Pb-210 remains.

We will use half-life formula to solve our given problem.

$A=a\cdot (0.5)^{\frac{t}{h}}$, where,

A = Amount after t units of time,

a = Initial amount,

t = Time,

h = Half-life.

$25=100\cdot (0.5)^{\frac{t}{22}}$

$\frac{25}{100}=\frac{100\cdot (0.5)^{\frac{t}{22}}}{100}$

$0.25=(0.5)^{\frac{t}{22}}$

Take natural log of both sides:

$\text{ln}(0.25)=\text{ln}((0.5)^{\frac{t}{22}})$

Using natural log property $\text{ln}(a^b)=b\cdot \text{ln}(a)$, we will get:

$\text{ln}(0.25)=\frac{t}{22}*\text{ln}(0.5)$

$\frac{\text{ln}(0.25)}{\text{ln}(0.5)}=\frac{t*\text{ln}(0.5)}{22*\text{ln}(0.5)}$

$\frac{-1.38629436}{-0.69314718}=\frac{t}{22}$

$2=\frac{t}{22}$

$2*22=\frac{t}{22}*22$

$44=t$

Therefore, the animal has been deceased to 44 years.

Answer 2

This is an easy half-life calculation.  (Most of these are NOT easy).
If 25% remains of an original amount of 100%, then the time elapsed is precisely 44 years.  (2 half-lives)
In 22 years (one half-life), it would be 50%
In 44 years (two half-lives), it would be 25%
In 66 years (three half-lives), it would be 12.5%
In 88 years (four half-lives), it would be 6.25%