Question

The number of bacteria present in a culture at any time, t hours, can be modelled by the equation N=N0ekt.
a) if the original number is doubled in 3 hours, find k correct to 2 decimal places.
b)write the equation substituting the value of k.
c)find the original number of bacteria if there were 2500 bacteria after 4 hours. give the answer correct to the nearest thousand.
d)write the equation substituting your value for the original population.
e)Find the number of bacteria present after 8 hours. Give the answer correct to the nearest thousand.

Answer 1

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Answer 2

\[\Large\rm N(t)=N_o e^{kt}\]When t=3, the amount that we started with, \(\Large\rm N_o\), has doubled,\[\Large\rm 2N_o=N_o e^{3k}\]

Answer 3

Solve for k.

Answer 4

Understand how to do that?

Answer 5

nope

Answer 6

could you show me how to solve for k, please

Answer 7

If we divide each side by \(\Large\rm N_o\) we get,\[\Large\rm 2=e^{3k}\]Taking the natural log of each side gives us,\[\Large\rm \ln2=\ln\left(e^{3k}\right)\]Applying a logarithm rule allows us to bring the 3k down in front,\[\Large\rm \ln2=3k\ln\left(e\right)\]Then isolate your k,\[\Large\rm \frac{\ln2}{3\ln e}=k\]

Answer 8

Toss it into the calculator, what do you get? :U

Answer 9

how can you sub e into the calculator??

Answer 10

mine comes as e^-

Answer 11

You should be able to simplify \(\Large\rm \ln e\) before using your calculator.
Whenever you have a log where the `base` matches the `contents` of the log, the result is 1.\[\Large\rm \ln e=1\]So our k is really,\[\Large\rm \frac{\ln2}{3}=k\]

Answer 12

ohh okay

Answer 13

If you want to do it that way, then you would type it like this:\[\Large\rm \ln(e\text{^}(1))\]

Answer 14

a 1 for the power.

Answer 15

e^1 = e

Answer 16

okie, I get 0.2310

Answer 17

Mmm ok good.
So we round it to 2 decimals then sub it back into our equation.\[\LARGE\rm N(t)=N_o e^{(.23t)}\]

Answer 18

c) Find the `original number of bacteria` if there were 2500 bacteria after 4 hours. give the answer correct to the nearest thousand.

Do you understand what part c) is asking for?
Given that at t=4 hours, N=2500,
Find \(\Large\rm N_o\), the original number of bacteria.

Answer 19

\[\Large\rm N(4)=2500\]Plugging in this information gives us,\[\LARGE\rm 2500=N_o e^{(.23\cdot4)}\]Understand what I did there?

Answer 20

yes :)

Answer 21

Hmm so to solve for \(\Large\rm N_o\) just requires some basic division.
What do you get? :U