Question

Molecules of a toxic chemical eventually decompose into inert substances. Suppose the decomposition time is exponentially distributed with a mean of 1/lambda. The half-life of such a persistent poison is that time beyond which the probability is .50 that a particular molecule will remain toxic. Find the half-life for chemicals whose molecules have an average decomposition time of

Answer 1

Answer:

a)4.15 years

b)22.18 years

c)188.53 years

d)3027.66 years

Step-by-step explanation:

Formula : $P(X\leq t)=1-e^{\frac{-t}{\lambda}}$

So, $P(X>t)=1-(1-e^{\frac{-t}{\lambda}})$

$P(X>t)=e^{\frac{-t}{\lambda}}$

We are given that The half-life of such a persistent poison is that time beyond which the probability is .50 that a particular molecule will remain toxic i.e. P(X>t)=0.5

So, $e^{\frac{-t}{\lambda}}=0.5$

Taking natural log both sides

$ln(e^{\frac{-t}{\lambda}})=ln(0.5)\\\frac{-t}{\lambda}=ln(0.5)\\t=-\lambda ln(0.5)$

a)

$\lambda =6$

t=-(6) ln(0.5)=4.15 years

b)

$\lambda =32$

t=-(32) ln(0.5)=22.18 years

c)

$\lambda =272$

t=-(272) ln(0.5)=188.53 years

d)

$\lambda =4368$

t=-(4368) ln(0.5)=3027.66 years