Question

Suppose a culture of bacteria starts with 9,000 bacteria. After one hour, the count is 10,000. Find an exponential equation that models the number of bacteria in x 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

easy way or hard way?

Answer 2

Easy! :)

Answer 3

ok
you start with 9000, so that will be the number out front
then \[\frac{10,000}{9,000}=\frac{10}{9}\] one one easy model is \[P(x)=9000\times \left(\frac{10}{9}\right)^x\]

Answer 4

it is harder if you want to make it look like \[\huge P(x)=P_0e^{kx}\]

Answer 5

I see, I actually think my teacher wants us to use the harder way :( would you mind explaining it?

Answer 6

ok no problem
same \(P_0\) as before, what you start with

Answer 7

but to find \(k\) requires a little work (not much)
you know in one hour it goes from \(9,000\) to \(10,000\)so set \[9000\times e^{1k}=10,000\] and solve for \(k\)

Answer 8

you know how to do that (hint, it requires logs)

Answer 9

Oh! Gotcha! Thank you so much!

Answer 10

you know how to solve that for \(k\)?

Answer 11

I think I do using logs right?

Answer 12

yes, divide both sides by \(9000\) first to get \[e^k=\frac{10}{9}\]
the write in logarithmic form
you will need a calculator

Answer 13

Perfect! Thanks again!

Answer 14

yw

Answer 15

Here are some formulas you might find helpful.

Answer 16

Oh thank you so much!