## Exponential Growth and Decay

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### Exponential Growth and Decay

So I have been given a couple of questions from my tutor however I have no idea where to start, I'm not asking for the answers just a push in the right direction.

1)

270 Vibrio cholerae cells were inoculated into 10 ml of a liquid culture and incubated for 9 hours. At the end of this time the cells (in a volume of 100 microlitres) were spread onto agar plates and the colony forming units were enumerated after overnight incubation by counting the colonies on plates from different dilutions: e-5 = 643; e-6 = 75; e-7 = 6

Assuming that the cells multiplied by binary fission and the culture was in synchronized exponential growth throughout the incubation, how many

2)

500 microlitres of a solution of a DNA fragment labelled with 32P with an initial activity of 0.4 kBq in 10 ml was needed for an experiment 9 days later. Calculate the radioactivity of the sample used in the experiment (T = half-life)

3)

Barium-140 has a half light of 13 days. How many decay half-lives does a sample of barium-140 undergo in 6 weeks?

What fraction of the sample would remain after 6 weeks?

1)

270 Vibrio cholerae cells were inoculated into 10 ml of a liquid culture and incubated for 9 hours. At the end of this time the cells (in a volume of 100 microlitres) were spread onto agar plates and the colony forming units were enumerated after overnight incubation by counting the colonies on plates from different dilutions: e-5 = 643; e-6 = 75; e-7 = 6

Assuming that the cells multiplied by binary fission and the culture was in synchronized exponential growth throughout the incubation, how many

**full**generations did the culture undergo and what was the generation time.**I guessed that you can only use one of the counts from the dilutions to perform the calculations and the number should be between 30 and 300 so I guess i'm using the 10^-6**2)

500 microlitres of a solution of a DNA fragment labelled with 32P with an initial activity of 0.4 kBq in 10 ml was needed for an experiment 9 days later. Calculate the radioactivity of the sample used in the experiment (T = half-life)

**(I think I first need to work out the amount of radioactivity in the sample to be used in the experiment, but couldn't work that one out)**3)

Barium-140 has a half light of 13 days. How many decay half-lives does a sample of barium-140 undergo in 6 weeks?

What fraction of the sample would remain after 6 weeks?

**(Confusing question, I guessed barium-140 would go through 3.23 half-lives and that it would be around 1/8th of the sample left over, but exact number eludes me)**
Last edited by Chemhalp on Wed Apr 23, 2008 7:59 pm, edited 1 time in total.

### Re: Exponential Growth and Decay

So for question three,

42 days / 13 day half life = 3.23

ergo

0.5^3.23 = 0.106579361, which is fraction of the sample after the 6 weeks?

42 days / 13 day half life = 3.23

ergo

0.5^3.23 = 0.106579361, which is fraction of the sample after the 6 weeks?

http://en.wikipedia.org/wiki/Exponential_growth

If you manipulate the formula a little, its basically what you'll need for exponential stuff.

If you manipulate the formula a little, its basically what you'll need for exponential stuff.

Living one day at a time;

Enjoying one moment at a time;

Accepting hardships as the pathway to peace;

~Niebuhr

Enjoying one moment at a time;

Accepting hardships as the pathway to peace;

~Niebuhr

### Re: Exponential Growth and Decay

Ok just had a look at question 1 again...

270 cells in 10ml then 100 microlitres taken

So... 10000/100 = 100 so 270/100 = 2.7

So... starting with 27 and ending with 75 is what I have right?

original stuff * (1/2)^n = stuff left

as you said above

So... 2.7 * 0.5^(n) = 75

75/2.7 = 0.5^(n)

But that gives me like 0.5^n = 27 which can't be...

So then I got thinking that its at dilution e-6 so something else should have been done -_-

Any hints?

270 cells in 10ml then 100 microlitres taken

So... 10000/100 = 100 so 270/100 = 2.7

So... starting with 27 and ending with 75 is what I have right?

original stuff * (1/2)^n = stuff left

as you said above

So... 2.7 * 0.5^(n) = 75

75/2.7 = 0.5^(n)

But that gives me like 0.5^n = 27 which can't be...

So then I got thinking that its at dilution e-6 so something else should have been done -_-

Any hints?

I'm not quite sure but from the way you wrote it, it seems like this is how it looks

You have a dilution with "x" number of colonies and it multiplied "n" times

x* 2^n=6

for the second dilution

10x * 2^n = 75

from third

100x * 2^n =643

But it's impossible to solve for the x or n because all 3 equations are linearly dependent...so I'm not sure what additional information I'm missing...

You have a dilution with "x" number of colonies and it multiplied "n" times

x* 2^n=6

for the second dilution

10x * 2^n = 75

from third

100x * 2^n =643

But it's impossible to solve for the x or n because all 3 equations are linearly dependent...so I'm not sure what additional information I'm missing...

Living one day at a time;

Enjoying one moment at a time;

Accepting hardships as the pathway to peace;

~Niebuhr

Enjoying one moment at a time;

Accepting hardships as the pathway to peace;

~Niebuhr

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