Question

The count in a bateria culture was 400 after 10 minutes and 1400 after 35 minutes. Assuming the count grows exponentially,

What was the initial size of the culture?

Find the doubling period.

Find the population after 110 minutes.

When will the population reach 13000

Answer 1

y = a x e^(kt) (equation for exponential growth)
\[y = a \times e^{k \times t}\]
so in this: a = initial number of bacteria
k = growth rate
t = time
y = number of bacteria now

Answer 2

y = a x e^(kt)
1400 = a x e^35k
so a = 1400 / e^35k

sub that into
y = a x e^(kt)
400 = a x e^(k x 10)
400 = (1400 / e^35k) x e^(kt)\[400 = \frac{ 1400 }{ e^{35k} } \times e^{10k}\]
\[400 = \frac{ 1400e^{10k} }{ e^{35k} } \]
\[400 = 1400e^{10k-35k} \]
\[400 = 1400e^{-25k} \]
\[400 = \frac {1400}{e^{25k}} \]
\[400e^{25k} = 1400 \]
\[e^{25k} = \frac {1400}{400} \]\[e^{25k} = \frac {7}{2} = 3.5 \]

Answer 3

\[e^{25k} = 3.5 \]
\[\ln (e^{25k}) = \ln (3.5)\]
\[25k = \ln (3.5)\]
\[k = \frac {\ln (3.5)}{25}\]

Answer 4

so now u know k, solve for a...
use the value u find for a and k to solve parts b, c, and d

Answer 5

just to make sure....k=.05011 right?

Answer 6

yep

Answer 7

what did you get for a if you don't mind me asking?

Answer 8

I got 242

Answer 9

bang on!, nice, me too