Question #ab040

A radioactive isotope's half-life tells you the time needed for an initial sample of said isotope to be halved.

More specifically, an initial sample of a radioactive element will be halved for every half-life that passes. If you start at #t=0# and have #t_"1/2"# as the isotope's half-life, then an initial sample will change like this

#100% -> t = 0#

#50% -> t = t_"1/2"#

#25% -> t = 2 * t_"1/2"#

#12.5% -> t = 3 * t_"1/2"#

#6.25% -> t = 4 * t_"1/2"#

This is equivalent to saying that

#"what you have" = "what you started with"/2^n" "#, where

#n# - the number of half-lives that passed.

In your case, you start with a sample of 100 mg. In order for the sample to be reduced to 25 mg, you need to have

#"100 mg" -> t = 0#

#"50 mg" -> t = t_"1/2"#

#"25 mg" -> t = 2 * t_"1/2"#

Your remaining sample is now a quarter the size it was in the beginning, which can only mean that two half-lives have passed

#"25 mg" = "100 mg"/2^n#

#2^n = (100color(red)(cancel(color(black)("mg"))))/(25color(red)(cancel(color(black)("mg")))) = 4 implies n = color(green)(2)#

The time that passed is thus equal to

#t = 2 * t_"1/2" = 2 * "5760 years" = color(green)("11520 years")#