Carbon-14 has a half-life of 5770 years. If a fossil is 23,080 years old and it has 3kg of Carbon-14, how much carbon-14 did it originally have?
A radioactive isotope's nuclear half-life tells you how much time must pass in order for a sample of this isotope to the reduced to half of its initial size.
In essence, any sample you start with will be halved with every passing of a half-life. So if you start with a mass
#A_0 * 1/2 = A_0/2 ->#after one half-life
#A_0/2 * 1/2 = A_0/4 ->#after two half-lives
#A_0/4 * 1/2 = A_0/8 ->#after three half-lives
and so on. You can thus find a relationship between how many half-lives have passed in a given period of time and how much of your initial sample is left undecayed
#color(blue)(A = A_0 * 1/2^n)" "#, where
In your case, you know that the fossil is
This means that you determine how many half-lives have passed in this time period
#n = (23080 color(red)(cancel(color(black)("years"))))/(5770color(red)(cancel(color(black)("years")))) = 4#
So, you know that
#A = A_0 * 1/2^n implies A_0 = A * 2^n#
#A_0 = "3.0 kg" * 2^4 = color(green)("48 kg")#