the number N of bacteria in a culture is modeled by N=250e^kt, where t is the time in hours. if N=280 when t=10, estimate the time requires for the population to double in size.
i suggest you first find what is k
can you do that?
i tried and i get 0.11 but im nto sure if its right
when i plugged it back in tot he equaton again it turned out to be 7.33 hours, but thats not the right answer D:
you have \[\ln N = kt\] right..so k is \[\frac{\ln N}{t} = k\] is this what you did?
wait...
divide both sides by 250 first... \[\frac{N}{250} = e^{kt}\] then take ln \[\ln (\frac{N}{250}) = kt\] divide by t \[\frac{\ln (\frac{N}{250})}{t} = k\] is this what you did?
i thoguht it was 280=250e^k(10)?
yup it is...
that's also right...but you will still end up doing what i did
\[280 = 250e^{10k}\] \[\frac{280}{250} = e^{10k}\] \[\ln (\frac{280}{250}) = 10k\] \[\frac{\ln(\frac{280}{250})}{10} = k\] see?
so is that what you did to find k?
yep
i got 0.11
not 0.011?
uh oh... that's what happened!! oh gosh ahah thank you
no wonder the answer came out wrong
\[\ln (\frac{280}{250}) = 0.11\] you still need to divide by 10
wait so k isnt =0.011?
k is 0.011
and then my new equation is 560=250e^0.011t right?
would you like to know a quick formula on how to find the time when this thing will double?
yes please
use this formula \[\frac{\ln 2}{k}\]
would you like to know the derivatin behind this?
yes
okay..the formula for exponential growth is \[\ln (\frac{x}{x_o} )= kt\] x is the final population x_0 is the initial population t is the time k is the constant so let's call the time when population will double as t" (notice the double-apostrophe) so at t", final population will be twice the initial so \[x = 2x_o\] if i substitute this into the formula for exponential growth \[\ln(\frac{2x_o}{x_o}) = kt"\] \[\ln(\frac{2\cancel{x_o}}{\cancel{x_o}}) = kt"\] \[\ln 2 = kt\prime\] remember we want t' so we divide both sides by k \[t "= \frac{\ln 2}{k}\]
that's supposed to be \[\ln 2 = kt"\]
does that make sense?
you are good at explaining, thank you :D
haha thanks ^_^
i can also do it from where you were going a while ago since k = 0.011 the new equation would be 500 = 250e^0.011t see how 500 is twice 250? what you did wrong was what you multiplied by 2 was the population at time = 10 should've been the 250 this one will give you the same answer :)
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