if your smoke detector contains 0.10 mg of 241Am, how long will it take to decay 1.25 * 10^-2mg

Answer 1

I assume you mean how long will it take to decay from 0.10 mg to 1.25E-2 mg?

k = 0.693/t_1/2
ln(No/N) = kt
No = 0.10 mg
N = 1.25E-2 mg
k from above.
Substitute and solve for t. You will need to look up the half life of Am241.