Question

If we start with 8000 atoms of radium-226, how much would remain after 3,200 years?

Answer 1

`"2000 atoms"`

Explanation:

The problem doesn't provide you with the nuclear half-life of the radium-226 isotope, so you're going to have to do a bit of research on that.

You can find this isotope's half-life listed as

`t_"1/2" = "1600 years"`

https://en.wikipedia.org/wiki/Isotopes_of_radium

So, the half-life of a radioactive nuclide tells you how much time must pass in order for half of an initial sample of said nuclide to undergo radioactive decay.

In your case, the half-life of radium-226 tells you that a sample of this nuclide will need `"1600"` years to be reduced to half of its initial size.

Your starting sample contains `8000` atoms of radium-226, so you can say that after one half-life passes, you will be left with

`"8000 atoms" * 1/2 = "400 atoms " ->` after `1600` years

How about after two half-lives pass? The remaining sample will be halved again

`"4000 atoms" * 1/2 = "2000 atoms " ->` after `3200` years

Therefore, your initial sample of radium-226 will be down to

`"no. of atoms that remain undecayed" = color(green)(|bar(ul(color(white)(a/a)color(black)("2000 atoms")color(white)(a/a)|)))`