Question

A common inhabitant of human intestines is the bacterium Escherichia coli, named after the German pediatrician Theodor Escherich, who identified it in 1885. A cell of this bacterium in a nutrient-broth medium divides into two cells every 20 minutes. The initial population of a culture is 40 cells.
(a) Find the relative growth rate in hours.

Answer 1

What we are given initial # of cells = 40. doubles every 20 minutes.

Answer 2

how many are there after 1 hour? after 2 hours? after 3 hours?

Answer 3

or rather, after 3 periods, after 6 periods, after 9 periods .... 3 periods of 20min in an hour right?

Answer 4

no it just wants us to find the general relative growth rate

Answer 5

then youll have to define the terminology to see if my idea is on the right track or not

Answer 6

check if I did my work correct, one sec

Answer 7

\[y=40e ^{kt}\]
\[80=40e ^{k \frac{ 20 }{ 60 }}\]
\[2=e ^{k \frac{ 1 }{ 3 }}\]
\[\ln 2=\ln e ^{k \frac{ 1 }{ 3 }}\]
\[k=3\ln 2\]

Answer 8

is this calculus work?

Answer 9

according to my book k is the relative growth rate. we are using Calculus Early Transcedentals 8 edition. and yes

Answer 10

\[\frac{dP}{dt}=kP\]
\[k(\color{red}{rgr})=\frac{1}{P}\frac{dP}{dt}\]

Answer 11

yes

Answer 12

So what does it mean?

Answer 13

P(0min)= 40
P(20min)= 80
P(40min)= 160
P(60min) = 320
20'; 640
40'; 1280
60'; 2560

P(0 hours) = 40
P(1 hours) = 320
P(2 hours) = 2560

320 = 40 r , r = 8

2560 = 40 8^3 ?? ... yes

P(t) = 40 8^t in terms of hours, as opposed to 20minutes

Answer 14

the question is asking to solve the value of k

Answer 15

yeah, i was trying to work out a similar problem, but this might be a better approach.

Answer 16

some texts are easier to read thru than others fer sure :)

Answer 17

ah ok I was correct in my procedure then

Answer 18

your process does match with their method yes :)

Answer 19

thank you

Answer 20

oh also one mroe thing

Answer 21

if we want to find the rate of growth, we take the derivative of y=40e^(tln8) right?

Answer 22

P(t) = 40 8^t

P'(t) =40 ln(8) 8^t

gives us k=ln(8) = 3ln(2)

Answer 23

that was the other one i was looking at ....

finally clicked as to where to get k from :)

but the site is confusing ... which is why i asked for definitions of the terms to start with, ive been far removed from the content.

Answer 24

oh ok lol

Answer 25

it seems that k is the growth rate, ... and the relative growth rate?

Answer 26

no in the book it says k is the relative growth rate, I dont know what the difference is to be honest.

Answer 27

oh wait that just gave me the answer lol, dp/dt is the growth rate

Answer 28

then is sounds like dP/dt is the growth rate since it is the thing being divded by P

Answer 29

I dont know why but when im working alone, I can't seem to think thats why I like coming on here. It helps me learn haha

Answer 30

but isnt that just solving for k?

Answer 31

pfft ... getting wires crossed lol

Answer 32

no dP/dt is not solving for k, it is the rate at which the cells are doubling in terms of time

Answer 33

dP/dt is being called the growth rate.

but that is just another function:

dP/dt = k P

Answer 34

yes

Answer 35

so if we want to find derivative of P(t) we just take the product of k and P

Answer 36

P = 40 8^t based on t=hours

dP/dt = ln(8) 40 8^t

yes

Answer 37

can you show the steps you took to get 8 as the base plz?

Answer 38

already did ... its posted above

Answer 39

oh yea ok

Answer 40

Well thank you so much again

Answer 41

youre welcome, hopefully it becomes more clear :) good luck

Answer 42

It increases by four times
The function is h=4t