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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?

Guest Jul 20, 2017
 #1
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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.

Guest Jul 20, 2017
edited by Guest  Jul 20, 2017
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why a negative exponent? just wondering :)

Guest Jul 21, 2017
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Because the quantity is decreasing over time.

Melody  Jul 21, 2017
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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.

Alan  Jul 20, 2017
edited by Alan  Jul 20, 2017

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