SOLUTION: The count in a bateria culture was 600 after 20 minutes and 1600 after 30 minutes. What was the initial size of the culture? Find the doubling period. Find the populat

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Question 885246: The count in a bateria culture was 600 after 20 minutes and 1600 after 30 minutes.
What was the initial size of the culture?
Find the doubling period.
Find the population after 90 minutes.
When will the population reach 11000
Answer by josgarithmetic(39618) About Me  (Show Source):
You can put this solution on YOUR website!
You may be able to do this using hours instead of minutes.


I = initial count of bacteria
p = count of bacteria after time t
highlight_green%28p=Ie%5E%28kt%29%29, exponential growth formula

Interesting that we do not know the count of bacteria at time t=0.

ln%28p%29=ln%28Ie%5E%28kt%29%29
ln%28p%29=ln%28I%29%2Bln%28e%5E%28kt%29%29
ln%28p%29=ln%28I%29%2Bkt%2A1
highlight_green%28ln%28p%29=ln%28I%29%2Bkt%29, a LINEAR equation. The vertical axis intercept is ln%28I%29 and the slope is k. The vertical axis variable is ln%28p%29 and is a function of time t.

You have two data points to help in getting vertical intercept and slope.
20 & 600;
30 & 1600.
The way you must use them first is this way:
20 & ln(600)=6.3969
30 & ln(1600)=7.3778
Meaning the two points for the LINEAR form of this description are these:
(20, 6.3969) and (30, 7.3778).
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k or SLOPE
k=%287.3778-6.3969%29%2F%2830-20%29
k=0.9809%2F10
k=0.09809
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Vertical Intercept
ln%28p%29=ln%28I%29%2Bkt
ln%28I%29=ln%28p%29-kt
Pick either point to finish finding ln(I).
ln%28I%29=7.3778-0.09809%2A30
ln%28I%29=4.4351
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Antilog, base e, for 4.4351 will be I.
I=84, more sensible than saying 84.36...
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Build back the exponential growth equation using the now found values:
highlight%28p=84%2Ae%5E%280.09809t%29%29.
You can answer whatever questions you want for this fictional or factual bacteria culture.

Find the doubling period.
Just let I=1 and p=2. Now find t.

Find the population after 90 minutes.
Obviously, just let t=90...

When will the population reach 11000
Let p=11000 and solve for t. There are clues how to do these i some of the steps I showed.