Question

Carbon-14 is a radioactive element with a half-life of about 5,730 years. Carbon-14 is said to decay exponentially. The decay rate is 0.000121. We start with one gram of carbon-14. We are interested in the time (years) it takes to decay carbon-14. (A) Find the value k such that P(x < k) = 0.6. (Round your answer to two decimal places.)

Answer 1

The require probability is defined by cumulative distribution function which is,

$P(x<k)=1-e^{-\lambda k}$

Where,

$P(x<k)=0.6$

$\lambda=0.000121$

Hence,

$P(x<k)=1-e^{-\lambda k}=0.6\Rightarrow e^{-\lambda k}=0.4$

$\Rightarrow e^{-\lambda k}=0.4\Rightarrow \text{taking natural log }\Rightarrow k=\frac{ln(2.5)}{\lambda}$

$\Rightarrow k=\frac{ln(2.5)}{\lambda}=\frac{0.916291}{0.000121}=7572.65068$

$\Rightarrow k=7572.65068 \text{ years}$