Differential Equation: decomposition of radium?
The radium in a piece of lead decomposes at a rate which is proportional to the amount present. If 10 percent of the radium decomposes in 200 years, what percent of the original amount of radium will be present in a piece of lead after 1000 years? Please solve by incorporating in the following formula: dx/x = -kt. Answer:59.05 percent.
- hmm, why isn't he providing any kind of work? - Well, I'm trying to teach myself coming from an Idiot's guide to Calculus which I just completed and am now trying ODEs when I found this being the very first question. Do I let x = 10% and k = years?
dR/dt = aR (first statement)
"try" (I know answer) R = Ro e-kt
dR/dt = -kRo e-kt = -kR yep a = -k
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R(200) = .9 Ro = Ro e(-k200)
.9 = e(-k200)
ln(.9) = -k 200
k = -ln(.9)/200
plug and chug; you now have k
% remaining after 1000 years is
100 * e(-k1000)
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another way to get it is you knock down to .9 after 200 years and you do that 5 times
(.9)^5 = 0.59049 ; yep agrees with prof
Idiot's Guide to Calculus AGD 2
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