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#### bergausstein

##### Active member

- Jul 30, 2013

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1. Radium decomposes at a rate proportional to the amount at any instant. In 100 years, 100mg of radium decomposes to 96mg. How many mg will be left after 200 years?

2. if a population of a town doubled in the past 25 years and the present population is 300,000 when will the town have a population of 800,000?

prob 1.

since 3mg of 100mg radium have decomposed over a period of 100 years this amount is 3% of the original amount.

$\frac{R_0-0.03R_0}{R_0}=\frac{R_0\,e^{k100}}{R_0}$

$\ln(1-0.03)=\ln(e^{k100})$

$\ln(1-0.03)=100k$

$k=\frac{\ln(1-0.03)}{100}$

when t=200

$R(200)=R_0\,e^{\ln(1-0.03)^2}$

$R(200)=100(0.9409)$

$R=94.09mg$ is this correct?

prob 2

$\frac{dP}{dt}=kP$

$P(t)=P_0\,e^{kt}$

when t=25; $P_0=2P_0$

$\frac{2P_0}{P_0}=\frac{P_0\,e^{25k}}{P_0}$

$2=e^{25k}$

$\ln2=25k$

$k=\frac{\ln2}{25}$

$\frac{dP}{dt}=kR$

$P(t)=P_0\,e^{kt}$

$P(t)=P_0\,e^{\frac{\ln2}{25}t}$

$800,000=300,000(2^{\frac{t}{25}}$

$\ln2.67=\frac{t}{25}\ln2$

$t= 35.42$years

is this correct?

2. if a population of a town doubled in the past 25 years and the present population is 300,000 when will the town have a population of 800,000?

prob 1.

since 3mg of 100mg radium have decomposed over a period of 100 years this amount is 3% of the original amount.

$\frac{R_0-0.03R_0}{R_0}=\frac{R_0\,e^{k100}}{R_0}$

$\ln(1-0.03)=\ln(e^{k100})$

$\ln(1-0.03)=100k$

$k=\frac{\ln(1-0.03)}{100}$

when t=200

$R(200)=R_0\,e^{\ln(1-0.03)^2}$

$R(200)=100(0.9409)$

$R=94.09mg$ is this correct?

prob 2

$\frac{dP}{dt}=kP$

$P(t)=P_0\,e^{kt}$

when t=25; $P_0=2P_0$

$\frac{2P_0}{P_0}=\frac{P_0\,e^{25k}}{P_0}$

$2=e^{25k}$

$\ln2=25k$

$k=\frac{\ln2}{25}$

$\frac{dP}{dt}=kR$

$P(t)=P_0\,e^{kt}$

$P(t)=P_0\,e^{\frac{\ln2}{25}t}$

$800,000=300,000(2^{\frac{t}{25}}$

$\ln2.67=\frac{t}{25}\ln2$

$t= 35.42$years

is this correct?

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