Question

I give medals!
Suppose a culture of bacteria starts with 8000 bacteria. After one hour, the count is 9000. Find an exponential equation that models the number of bacteria in hours. Keep at least 4 decimal places in your formula for rounded values. Find the doubling time of the bacteria. Round the doubling time to the nearest 0.01 hours.

Answer 1

simple way or hard way?

Answer 2

@satellite73 Whatever way you feel would be easiest to explain.

Answer 3

in one hour it goes from 8000 to 9000 so it increases by a factor of
\[\frac{9000}{8000}=\frac{9}{8}\] and therefore you can model it as
\[8000\times \left(\frac{9}{8}\right)^{t}\] where \(t\) is time in hours
pretty simple right?

Answer 4

if you need to make your teacher happy and use \(8000e^{kt}\) we can do that too, but it takes much longer

Answer 5

\[P(t)=P _{0}e ^{k*t}\] this is the correct equation right?

Answer 6

ok damn i guess we have to use that
i was hoping to get away with \(A=8000\times\left(\frac{9}{8}\right)^t\) but i guess not

Answer 7

it isn't that bad, just takes a bit longer to do
ready?

Answer 8

@satellite73 Yeah, that's what I was thinking earlier but I confused myself :o

Answer 9

ok so we know it is
\[8000e^{kt}\] we just don't know \(k\)

Answer 10

here is how we find it

\[A=8000e^{kt}\] if \(t=1\) then \(A=9000\) so we have to solve
\[9000=8000e^k\]for \(k\)

Answer 11

first divide by 8000 and get
\[\frac{9}{8}=e^k\] then take the log to get
\[k=\ln(\frac{9}{8})\] then use a calculator to find this number

Answer 12

I got 0.1178

Answer 13

So, the equation would be \[P(t)= 8,000e ^{0.1178t}\] right?! :)

Answer 14

let me check, looks good

Answer 15

yeah that is right

Answer 16

Alrighty! :) And for the second half I would just put into 2 for t right? Since that is doubled the time?

Answer 17

oh no
\(t\) is time, so if you put \(t=2\) you find out how many you have in two hours

Answer 18

Ah! I misread the problem! :o

Answer 19

to find the doubling time set
\[\large e^{.1187t}=2\] and solve for \(t\)

Answer 20

It's been a really long day! But thanks! I'll solve tat right now!

Answer 21

kk i'll check if you like

Answer 22

Please! I'll let you know when I am done

Answer 23

k

Answer 24

\[2=e ^{.1187*t}\]
\[\frac{ \ln2 }{ 0.1187 }=\frac{ 0.1187t }{ 0.1187 }\]
\[t=5.84 hours\]

Answer 25

that is what i got too
http://www.wolframalpha.com/input/?i=ln%282%29%2F.1187

Answer 26

Thanks so much! I really appreciate it! My brain is too fried from this week!

Answer 27

finals soon?

Answer 28

oh, and your welcome, good luck with this, looks like you are doing fine

Answer 29

Yeah, lots of them! Thanks good luck if you have any too! :)