Question

How long does it take for 90% of a given quantity of the radioactive element cobalt-60 to decay, given that itshalf-life is 5.3 years?

Answer 1

17.6 years (roughly)

Explanation:

ok so let's consider the amount of cobalt-60 to be:
`m` kg of cobalt.

We can model the decay of that cobalt given its half-life of
`5.3` as:

`f(t) = m((1)/(2))^{(t)/(5.3)}`

where
`t` is the time in years.

Now, for 90% of the cobalt to decay, we get the following equation:

`(m)/(10)=m* ((1)/(2))^(t)/(5.3)\\ \\ (1)/(10)=((1)/(2))^(t)/(5.3)`

and by using logarithms, we can find t.

`log((1)/(10))=log((1)/(2)^(t)/(5.3))\\ \\ log(1)-log(10)=(t)/(5.3) log((1)/(2))\\\\(log(1)=0)\\\\ -log(10)=(t)/(5.3) [log(1)-log(2)]\\\\(log[10]=1) \\\\`

`-1=((t)/(5.3) )* -log(2)\\\\\\(t)/(5.3)=(1)/(log(2))\\ \\t=(5.3)/(log(2))=17.6 years` (roughly)