Suppose a rock sample initially contains 250 grams of the radioactive substance
ID: 2855587 • Letter: S
Question
Suppose a rock sample initially contains 250 grams of the radioactive substance unobtanium, and that amount of unobtanium after t years is given by an exponential function of the form S(t)=Aekt . The half life of unobtanium is 29 years, which means it takes 29 years for the amount of the substance to decrease by half. (A) Find an equation for the exponential function S(t). (B) What percentage of unobtanium decays each year? (C) How long will it take before the rock sample contains only 6 grams of unobtanium?
Explanation / Answer
A) S(t)=Aekt
initially contains 250 grams ==> at t = 0 , S(0) = 250
==> 250 = Aek(0)
==> A = 250
half life is 29 years
==> at t = 29 , S(29) = 125 grams
==> S(29) = 250ek(29)
==> 125 = 250ek(29)
==> e29k = 1/2
Apply natural log on both sides
==> ln e29k = ln (1/2)
==> 29k = ln (1/2)
==> k = (1/29) ln(1/2) = -0.0239
==> S(t) = 250e-0.0239t
B) S(1) = 250e-0.0239(1) = 244.0958
[S(0) - S(1)]/S(0) *100 = [250 - 244.0958]/250 *100
= 2.362
Hence 2.362% decays every year
C) S(t) = 250e-0.0239t
S(t) = 6
==> 6 = 250e-0.0239t
==> e-0.0239t = 6/250
apply natural log on both sides
==> ln e-0.0239t = ln(6/250)
==> -0.0239t = ln(6/250)
==> -0.0239t = -3.7297
==> t = 3.7297/0.0239 = 156.05 years
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