Question

APPLICATION OF FIRST ORDER D.E. (EXPONENTIAL DECAY)

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: 82.95%

i was able to arrive with the answer but i had to "cheat" to get it. can someone please show me how to work this problem? i may have done something wrong

Answer 1

@Valpey ?

Answer 2

first order decay

Answer 3

okay here's what i did
\[\ln (\frac{x_o}{0.012x_o}) = 25k\]
is that right?

Answer 4

i wouldn't start like that

Answer 5

then what

Answer 6

\[X(t)=X_0e^{-k t}\]\[\frac{X(t)}{X_0}=e^{-k t}\]\[\ln\frac{X(t)}{X_0}=-kt\]\[\ln\frac{X_0}{X(t)}=kt\]

Answer 7

\[k=\frac1{t}\ln\frac{X_0}{X_t}\]\[k=\frac1{25\times 10^6}\ln\frac{1}{0.012}\]

Answer 8

xt is?

Answer 9

\(X_t\) should be \(X(t)\)
sorry

Answer 10

so have for found \(k\)

Answer 11

so it's \[\ln (\frac{x_o}{x(t)})?\]

Answer 12

what you just wrote is \(=kt\)

Answer 13

should it not?

Answer 14

so in a way what you did is just the same as i did right? \[\large \frac{\ln (\frac{x_o}{0.012x_o})}{25} = k\]
correct?

Answer 15

@UnkleRhaukus

Answer 16

yeah that is right if your working in millions of years

Answer 17

yup...so k = 0.1769 /million years

Answer 18

what do i do next?

Answer 19

now just use the first equation i wrote ,up the page
to find how much DNA will remain after 10million years

Answer 20

a take this away from 100%

Answer 21

that's actually my question...why do i use \[\ln (\frac{x_o}{x_o - ax_o})\] where a is the percent.

when in the other one i just used \[\ln (\frac{x_o}{0.012x_o})\]
what's the differece?

Answer 22

where did you get this from/\[\ln \left(\frac{x_o}{x_o - ax_o}\right)\]

Answer 23

\[X(t)=X_0e^{-k t}\]
\[X(t)/X_0=e^{-0.18t}\]
\[X(t)\%=e^{-1.8}\]

Answer 24

percentage lost:
\[1-X(t)\%\]

Answer 25

yes @UnkleRhaukus my question is how do i know when to do 1 - x(t) or just simply x(t)

Answer 26

x(t) is how much is remaining
1-x(t) is how much has decayed

Answer 27

OHHHH I SEE

Answer 28

http://www.wolframalpha.com/input/?i=100-100*e^%28-0.176914*10%29

Answer 29

eightee-free