Americium-241 is used in some smoke detectors. it is an alpha emitter with a half-life of 432 years. how long will it take in years for 45.0 % of an am-241 sample to decay?

The amount of substance at a certain time t, At, with half-life of h is determined by the equation,

At = (Ao) (0.5) ^ (t/h)

where Ao is the original amount.

From the given,

At/Ao = 0.45 = (0.5) ^ (t/432)

The value of t from the equation is 497.67 years.