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Evgesh-ka [11]
3 years ago
10

Molecules of a toxic chemical eventually decompose into inert substances. Suppose the decomposition time is exponentially distri

buted 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 (a) 6 yearshalflife= ? years(b) 40 yearshalflife= years(c) 344 yearshalflife= ? years(d) 5584 yearshalflife= ? years
Mathematics
1 answer:
koban [17]3 years ago
5 0

Answer:

a) 4.16 years

b) 27.73 years

c) 238.44 years

d) 3,870.53 years

Step-by-step explanation:

Let X be the random variable that measures the decomposition time.

a)

\bf \lambda =6

In this case, since the decomposition time is exponentially distributed with a mean of 1/6, we have

\bf P(X\leq t)=1-e^{-t/6}\Rightarrow P(X>t)=1-(1-e^{-t/6})=e^{-t/6}

and we must find a t such that P(x>t)=0.5.

\bf P(X>t)=0.5\Rightarrow e^{-t/6}=0.5\Rightarrow -t/6=ln(0.5)\Rightarrow t=-6ln(0.5)=4.16\;years

b)

\bf \lambda =40

\bf P(X>t)=0.5\Rightarrow e^{-t/40}=0.5\Rightarrow -t/40=ln(0.5)\Rightarrow t=-40ln(0.5)=27.73\;years

c)

\bf \lambda =344

\bf P(X>t)=0.5\Rightarrow e^{-t/344}=0.5\Rightarrow -t/344=ln(0.5)\Rightarrow t=-344ln(0.5)=238.44\;years

d)

\bf \lambda =5584

\bf P(X>t)=0.5\Rightarrow e^{-t/5584}=0.5\Rightarrow -t/5584=ln(0.5)\Rightarrow t=-5584ln(0.5)=3870.53\;years

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