Question

scientific studies show that DNA deteriorates over time. if excavations and diggings of fossils indicate that only 1.2% of DNA survive after 25 million years, what percent of an original sample would have been lost after 10 million years?

Answer 1

apply exponential decay formula
\[\huge x=ce^{−kt}\]

isolate t
\[\huge \ln(\frac xc)= −kt\]

\[\huge \implies −\frac{\ln(\frac xc)}k=t\]

simplify
\[\huge \frac{ln(\frac cx)}k=t\]


after 2.5 M years, 1.2% of DNA survived

let c be the original value. 1.2% is 0.012c

\[\huge \frac {\ln(\frac c{0.012c})}k=2.5\]

solve for k
\[\huge \implies k=\frac{\ln(\frac c{0.012c})}{2.5}\]

simplify
\[\huge k= \frac{\ln(83.3\overline 3)}{2.5}\]

solve for k
\[\huge \implies k=1.7691\]

Answer 2

now...solve for the percentage remaining after 10 million years
\[\huge \implies \frac{\ln (\frac c{ac})}{0.17691} = 10\]
where:
a = percentage

now...solve for a
\[\huge \implies \ln (\frac c{ac}) = 1.7691\]
\[\huge \frac c{ac} = 5.8656\]
\[\huge \frac 1a = 5.8656\]
\[\huge \frac 1{5.8656} = a\]
\[\huge a = 0.1705\]
this is the amount remaining.

to solve the amount lost, subtract that from 1
\[\huge \implies 1 - 0.1705 = 0.8295\]
so.. 82.95% was lost

Answer 3

got it @Jlastino ?

Answer 4

yep...thanks!