Question

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.

Answer 1

i suggest you first find what is k

Answer 2

can you do that?

Answer 3

i tried and i get 0.11 but im nto sure if its right

Answer 4

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:

Answer 5

you have \[\ln N = kt\]
right..so k is
\[\frac{\ln N}{t} = k\]
is this what you did?

Answer 6

wait...

Answer 7

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?

Answer 8

i thoguht it was 280=250e^k(10)?

Answer 9

yup it is...

Answer 10

that's also right...but you will still end up doing what i did

Answer 11

\[280 = 250e^{10k}\]
\[\frac{280}{250} = e^{10k}\]
\[\ln (\frac{280}{250}) = 10k\]
\[\frac{\ln(\frac{280}{250})}{10} = k\]
see?

Answer 12

so is that what you did to find k?

Answer 13

yep

Answer 14

i got 0.11

Answer 15

not 0.011?

Answer 16

uh oh... that's what happened!! oh gosh ahah thank you

Answer 17

no wonder the answer came out wrong

Answer 18

\[\ln (\frac{280}{250}) = 0.11\]
you still need to divide by 10

Answer 19

wait so k isnt =0.011?

Answer 20

k is 0.011

Answer 21

and then my new equation is 560=250e^0.011t right?

Answer 22

would you like to know a quick formula on how to find the time when this thing will double?

Answer 23

yes please

Answer 24

use this formula \[\frac{\ln 2}{k}\]

Answer 25

would you like to know the derivatin behind this?

Answer 26

yes

Answer 27

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
\[t "= \frac{\ln 2}{k}\]

Answer 28

that's supposed to be \[\ln 2 = kt"\]

Answer 29

does that make sense?

Answer 30

you are good at explaining, thank you :D

Answer 31

haha thanks ^_^

Answer 32

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 :)