# Question #e3336

##### 1 Answer

#### Explanation:

The **half-life** of a radioactive nuclide, **half** of a given sample to undergo radioactive decay.

So if you take **initial mass** of carbon-14, you can say that this sample will be reduced to

#A_0 * 1/2 = A_0/2 = A_0/2^color(red)(1) -># after#color(red)(1)# half-life#A_0/2 * 1/2 = A_0/4 = A_0/2^color(red)(2) -># after#color(red)(2)# half-lives#A_0/4 * 1/2 = A_0/8 = A_0/2^color(red)(3) -># after#color(red)(3)# half-lives

#vdots#

and so on.

Now, let's say that **remains undecayed** after a period of time

#A_t = A_0 * (1/2)^color(red)(n)#

with

#color(red)(n) = t/t_"1/2"#

Here **number of half-lives** that pass in the given period of time

In your case, the sample has an initial mass of **years**, so you can say that after a period of time

#A_t = "52 mg" * (1/2)^(t/"5730 years")#

To find the amount of carbon-14 that remains undecayed after **years**, simply plug in this value into the above equation.

You will have

#color(red)(n) = ("10,000" color(red)(cancel(color(black)("years"))))/(5730color(red)(cancel(color(black)("years"))))#

and

#A_t = "52 mg" * (1/2)^("10,000"/5730) = color(darkgreen)(ul(color(black)("15.5 mg")))#

The answer is rounded to the nearest tenth.