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 63 cells.


Find the rate of growth after 8 hours. (Round your answer to three decimal places.)

Answer 1

I know that
\[p(t) = 63 e^{2.07944*t}\]

And I know that there are 1056964608 cells after 8 hours.


This is driving me nuts. I've been trying to figure out what rate of growth means. Relative rate of growth is easy, but this... so frustrated....

Answer 2

What I mean when I say, "i know that.." from above, is that I figured it out.
Although I am certain my math is correct. So the above values are right.

Answer 3

i got the other one i believe in the end lol

Answer 4

"medium divides into two cells every 20 minutes. The initial population of a culture is 63"

20min = 1/3 hr

3 ln(2) = r then

so after 8 hours that should be:

A = 63 e^(8*3ln(2)) = 1.05696461 × 10^9

Answer 5

thats not the rate of growth tho, thats the population size

Answer 6

the rate of growth is the derivative of the function when t=8 i believe

Answer 7

A = 63 e^(3ln(2)*t)

A' = 3ln(2)*63 e^(3ln(2)*t)

Answer 8

there would have been a t' but t'=1 with respect to itself so we tend to toss it out

Answer 9

2197896113.96054....

http://www.wolframalpha.com/input/?i=3ln%282%29*63+e^%283ln%282%29*8%29

Answer 10

hopefully that makes sense ....

Answer 11

No idea what you did here, "20min = 1/3 hr 3 ln(2) = r then so after 8 hours that should be:"

I just assumed t was in hours. Made it much simpler.

Answer 12

If i understand correctly, you did not assume t was in hours, but rather t=20min?

If so, how did you get 3 ln(2)=r?

Answer 13

Nevermind ^^. I realized you weren't calculating rate of growth.

Answer 14

i tapped into a trick I recall that continuous things have a doubling effect every ln(2)/k where k is either the rate of the time depending on which one you need,

ln(2)/r = time

ln(2)/t = rate since 1/3 of an hour is the time it takes to double; the rate itself is 3 ln(2)

Answer 15

the rate OR the time ... accursed typos

Answer 16

I think you're approaching this from a different way than I am supposed to. It might eventually lead to the same conclusion, but.. here let me paste the ENTIRE question.

Answer 17

it might be (rt)' in the derivative ... but im pretty sure since its a constant that it kinda takes care of itself. k

Answer 18

There were 4 parts to this question. I got 3 of them correct.

(a) Find the relative growth rate. (Assume t is measured in hours.)
(b) Find an expression for the number of cells after t hours.
(c) Find the number of cells after 8 hours
(d) Find the rate of growth after 8 hours. (Round your answer to three decimal places.)
(e) When will the population reach 20,000 cells? (Round your answer to two decimal places.)

D is the only one I cannot get. I don't think we are supposed to use what you referred to as 'the doubling rule'. Which i think is just the 'Rule of 72' : ln(2)/ln(1+r)

Answer 19

Don't work out a, b, c, e above by the way. Web assign said I got them right so no need for checking..

And i meant to say there are 5 parts to this question, i got 4 correct. heh

Answer 20

2P = Pe^(rt)

2P/P = e^(rt)

2 = e^(rt)

ln(2) = ln(e^(rt))

ln(2) = rt

ln(2)/t = r

t, in hours is 1/3

ln(2)/(1/3) = r = 3 ln(2)

Answer 21

thats just the algebra

Answer 22

as you can see, the doubling has nothing to do with the initial size; it is only affected by time

Answer 23

or how fast, the rate ....

Answer 24

Find the rate of growth after 8 hours.

How fast is the population growing after 8 hours; thats a rate of change question that says we take the derivative

Answer 25

Okay, see what you did.
"t, in hours is 1/3 ln(2)/(1/3) = r = 3 ln(2)"

the question D is asking for after 8 hours. So, i plug in 8 rather than 1/3 for t?

Answer 26

no, not that i am interpreting it. The initial equation for the growth tells us size for any given t in hours.

teh rate at which that growth occurs is the dervative of the size

Answer 27

the size is:

\[A(x) = 63e^{3ln(2)\ t}\]
\[A'(x) = [3ln(2)\ t]'*63e^{3ln(2)\ t}\]
\[A'(x) = 3ln(2)*63e^{3ln(2)\ t}\]

Answer 28

the rate of growth is then A'(x)

Answer 29

lol ... i used an x :) A(t) and A'(t)

Answer 30

if you plug in an 8 instead of the 1/3 your implying that this thing doubles every 8 hours which is false

Answer 31

Okay yeah, i got that. Look above at the second comment.

I just wrote it differently. Instead of writing 3ln(2), i just had 2.0794415.

So all I needed to do was find p'(t) from the beginning?

God dammit!

Answer 32

yep

Answer 33

Thank you kind sir!

Answer 34

youre welcome, hope it helps ;)

Answer 35

Sigh, I'm having trouble taking the derivative.. Mind telling me where I'm screwing up in my steps?
\[A(x)=63e^{3\ln(2) t}\]
*product rule*

\[A'(x)= [63 * e^{3\ln(2)t}] + [e^{3\ln(2)t} *0]\]

So it comes back to the exact same thing? Since d/dx e^x = e^x???...

But I know this can't be right?

Answer 36

Oh wait, did you do chain rule? How do you use chain rule with e^3ln(2)t? Did you just bring down the entire exponents and take the derivative of them? Weird...

Answer 37

you learnt the rule for "e" a little off. Every function pops out a chain rule.

think of e^x as e^(f(x)) such the derivative of e^(f(x)) = e^(f(x))* f'(x)

Answer 38

derivatives always pop out an x'; but x' with respect to x = 1 so it tends to be discarded with no afterthought

Answer 39

d/dx [3ln(2)t = 3ln(2) according to your work above?
But shouldn't it just be 3/2. Since d/dx ln(x) = 1/x

Answer 40

3ln(2) is a constant; there are no variables in it to operate upon. think of it as [Ct]' ; which is just C

Answer 41

d\dx ln(4) = 0 since ln(4) is just a constant. When something doesnt change, its rate of change is 0

Answer 42

if your always 5 years old, you never grow up :)

Answer 43

Silly mistake, i get it. TY

Answer 44

3ln(2) = 2.079 ...; the derivative of 2.079... is just 0

Answer 45

;)

Answer 46

Hmmm, I just plugging in 8 for t into

A't =

, got 2197896114, and I'm getting that that is wrong.

The answer is in format:

[answer box here] billion cells per hour.

So do i divide by 8.. Don't know what to do.

Answer 47

2 197 896 114

that is about 2.2 billion