Julius K. answered • 12/02/19
Well-rounded scientist who specializes in Chemistry & Physics
The half life of an element is the time required for half of the mass of a sample to decay.
For example, after 28 years, only 30 mg of the original 60 mg sample would remain, after another 28 it would be 15 mg, and so on and so forth.
When the target mass isn't a nice fraction of the original (like it is here), it's easier to use an equation:
A = A0 x (1/2)t/h
Where A is the final amount (in this case, 26.4 mg), A0 is the amount you started with (60 mg), t is the total time that has passed, and h is the half life (28 years). As long as the half life and the elapsed time are the same units (years, minutes, days, etc.), it doesn't really matter what they are.
Plugging in the values we have, we get:
26.4 mg = 60 mg x (1/2)t/h
To solve the problem, we need to get t by itself. We can simplify:
(26.4 mg/60 mg) = (1/2)t/h
Then, we can use a logarithm rule to get
(26.4 mg/60 mg) = (2)-t/h
t = [(28 years) x ln(26.4/60)]/ ln(0.5)
t = ~33.16388799.... years
Which makes sense, since it should take longer than 1 half life to have less than 30 mg, but less than 2 since there's more than 15 mg.
