household smoke detectors contain 200 ug of a radioactive element that has a half-life of 432 years. to the nearest 10 ug, how much of this element will remain in a smoke detector after 20 years?

R =200 x 2^-(20/432)

R =200 x 2^-(1/21.6)

R =200 x 0.96841927.....

R =193.684 ug remains after 20 years.

why a negative exponent? just wondering :)

Because the quantity is decreasing over time.

N(t) = N(0)*e^{-kt} where k = ln(2)/tau tau is half-life.

k = ln(2)/432 ≈ 1.605*10^{-3} per year

N(20) = 200*e^{-1.605*10^(-3)*20} ug ≈ 193.7 ug = 190 ug to the nearest 10 ug.