Question

"A common inhabitant of human intestines is the bacterium Escherichia coli. A cell of this bacterium in a nutrient-broth medium divides into two cells every 20 minutes. The initial population of a culture is 65 cells."

I'm supposed to determine the rate of growth after 8 hours.

What I've got:
relative growth rate = k = ln(8)
expression for the number of cells after t hours = P(t) = 65e^(8ln(t))
rate of growth = dP/dt = 65*8^(t)*ln(8)

Thus, 65*8^(8)*ln(8) should be the rate of growth after 8 hours, unfortunately this seems to be wrong, why isn't it right?

Answer 1

why is growth rate \(\ln(8)\)?

Answer 2

doubles every 20 minutes, model is
\[P(t)=65\times 2^{\frac{t}{20}}\]

Answer 3

or if you need
\[65e^{kt}\] to find \(k\) solve
\[2=e^{20k}\] for \(k\)

Answer 4

you get
\[20k=\ln(2)\] and so
\[k=\frac{\ln(2)}{20}\]

Answer 5

Yep, which is the same as ln(8).

Answer 6

really?

Answer 7

Nope, sorry. ^_^

Answer 8

i am fairly sure that
\[\ln(2)<\ln(8)\] which means that
\[\frac{\ln(2)}{20}\] is way less

Answer 9

I solved mine differently, I just moved down stairs, lemme grab my notebook.

Answer 10

ok but hope the steps are clear. you know it doubles every twenty minutes so you solve
\[2=e^{20k}\] for k

Answer 11

once we get \(k=\frac{\ln(2)}{20}\) you want population in 8 hours, which is 480 minutes or 24 twenty minute intervals, so your final answer should be
\[65e^{24\ln(2)}\]

Answer 12

But it is ln(8)...
2 = e^(k/3)
k/3 = ln(2)
k = 3ln(2) = ln(8) [Answer in the back of the book is given in the same form as the second.]

Answer 13

Which makes sense since both = 2.07944154

Answer 14

ok i get it you are using hours as unit of time, i was using minutes

Answer 15

in any case the answer is the same, because we both end up with
\[65e^{24\ln(2)}\]

Answer 16

Um, not actually I got P(t) = 65e^(8ln(t)) which could also be represented as 65*8^t but I don't think that 65^(24ln(2)) is equivalent.

Answer 17

actually now i am a bit confused, because your question actually states
"I'm supposed to determine the rate of growth after 8 hours."
presumably the rate of growth is the same. doubles every 20 minutes

Answer 18

you want population in 8 hours. if you use hours as your unit of time, then you have
\[65e^{8\times \ln(8)}\] and that is the same as
\[65e^{24\ln(2)}\]

Answer 19

ln(8) is the relative growth rate which is different than the specific growth rate for a given time period. (Oh, you're right... natural logarithms are funny, aren't they...) I think it's supposed to be the derivative of the function at the specific point.

Answer 20

The point being 8 of course... since that's the time in hours we're looking for.

Answer 21

so the question is "what is the derivative of \(65e^{3\ln(2)t}\) at \(t=8\)"?

Answer 22

The question is what isn't dP/dy = ln(8)*8^t*65 at t = 8 correct, and why,,,

Answer 23

Because I followed the chain rule, don't have issue with derivatives or anything and it looks, for all intensive purposes, right. But I get this huge number and it's wrong.

Answer 24

derivative of \[65e^{3\ln(2)t}\] is
\[65\times 3\ln(2)e^{3\ln(2)t}\] by the chain rule

Answer 25

it is a huge number, so maybe i do not understand the question

Answer 26

At 2.27 hours there are 20,000 cells, so there should be a reasonable number of cells, but the rate of change couldn't possibly be 2267670593.7688146, it's an insane answer.

Answer 27

Yeah, I don't understand what I'm doing wrong, I guess it's an understanding issue.

Answer 28

do you know what the answer is supposed to be?

Answer 29

The specific question, word for word says... "Find the rate of growth after 8 hours. (Round your answer to three decimal places.)"

Answer 30

it seems like you have a solution, because you know \(k=\ln(8)\) or \(3\ln(2)\)

Answer 31

Yeah, it says that it's 15873694156.382

Answer 32

But I don't know where they're getting that from...

Answer 33

hmm let me use a calculator

Answer 34

ok i think i see it

Answer 35

nope, but closer

Answer 36

nope, i was wrong, damn

Answer 37

sorry i tried a bunch of different things to get that number and i cannot get it. i guess i do not understand the question. is there a similar example in the book to mimic?

Answer 38

Yeah, don't worry about it. I'm going to let my brain re-boot on it and I'll talk to my teacher about it in class later.

Answer 39

If I don't get it, that is...

Answer 40

good luck