Question

The population P of a bacteria culture is modeled by P = 4100e^(k*t), where t is the time in hours. If the population of the culture was 5800 after 40 hours, how long does it take for the population to double? Round to the nearest tenth of an hour.

Answer 1

u have
\[ \large P=4100e^{kt}. \]
for t=40 u have P=5800 then
\[ \large 5800=4100e^{40k} \]
\[ \large 1.4146=\frac{58}{41}=\frac{5800}{4100}=e^{40k} \]
\[ \large \ln(1.4146)=40k \]
\[ \large k=\frac{40}{\ln(1.4146)}=115.3167 \]

Answer 2

therefore
\[ \large P=4100e^{115.3167\times t} \]

Answer 3

finally
\[ \large 11600=5800\times2=4100e^{115.3167\times t} \]
\[ \large 2.8293=\frac{11600}{4100}=e^{115.3167\times t} \]
\[ \large \ln(2.8293)=115.3167t \]
\[ \large t=\frac{115.3167}{\ln(2.8293)}=110.8795 \]
so in approximately 111 hours, the population will double.

Answer 4

110 isn't one of the available answers. And I've tried re-working the problem and nothing really seems to work. Frustrating.

Answer 5

slight mistake in solving for k @helder_edwin

Answer 6

my bad sorry
\[ \large k=\frac{\ln(1.4146)}{40}=0.0087 \]

Answer 7

yeah i saw it

Answer 8

better

Answer 9

I managed to solve for k; k = 0.0087 ; not sure what to do then. x:

Answer 10

therefore
\[ \large P=4100e^{0.0087t} \]

Answer 11

you can use a shortcut actually \[\huge t = \frac{\ln 2}{k}\]
where t is the time it takes for the bacteria to double

Answer 12

so
\[ \large t=\frac{\ln(2.8293)}{0.0087}=119.9314 \]

Answer 13

Hm. That was the same answer I would get when I was trying to solve the problem, but that was wrong, too. :/

Answer 14

Answers are: A) 56.6 hours
B) 54.8 hours
C) 81.7 hours
D) 79.9 hours
E)8.9 hours

Answer 15

try using this \[t = \frac{ln 2}{0.00867}\]

Answer 16

you'll get one of those choices

Answer 17

maybe it is asking for the time it will take for the population to double from its starting point.

Answer 18

you were doubling the given value?

Answer 19

if you're going to double 5800 then why go through all the trouble of using logarithms...just double 40 hours and add 40....

Answer 20

so
\[ \large 8200=4100e^{0.0087t} \]
\[ \large \ln(2)=0.0087t \]
\[ \large t=\frac{\ln(2)}{0.0087}=79.6721 \]

Answer 21

Just confused right now. :(

Answer 22

@helder_edwin use 0.00867 as the constant...

Answer 23

in that case t=79.94777

Answer 24

yes. it shows the choices more

Answer 25

Thanks guys. I'll try to remember that. .:)