A bacteria culture will grow from 400 to 5000 in 1 1/4 hours.
a) Find the constant k for this bacteria using growth formula, y=ne^kt, and write the growth equation.
b)Use your equation to predict how much longer it will take the 5000 bacteria to become 15000 bacteria.

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- anonymous

\[y=ne^{kt}\]

- anonymous

n is beginning
y is end

- anonymous

Correct. Dont know where to start on this problem.

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- anonymous

so in beginning there was 400

- anonymous

so plug in 400 for n

- anonymous

plug in what for y?

- anonymous

5000?

- anonymous

yes

- anonymous

\[5000=400 e^{kt}\]

- anonymous

t represent time , what time do you see there

- anonymous

t would go in as 1.5

- anonymous

1 1/4= 1.25

- anonymous

I knew that.

- anonymous

let's go with that

- anonymous

k. so the equation is
\[5000=400e ^{k(1.25)}\]

- anonymous

yes

- anonymous

your job is to find k

- anonymous

Hmm...take ln of both sides and bring exponent down then solve for k?

- anonymous

exactly

- anonymous

\[\ln(5000)=k(1.25)\ln400e\]

- anonymous

not what I would do

- anonymous

How would you set it up?

- anonymous

I would divide both sides by 400 first

- anonymous

Ohh okay.

- anonymous

\[5000/400=e ^{k(1.25)}\]

- anonymous

now take ln of both sides

- anonymous

\[\ln(25/2)=k(1.25)lne\]

- anonymous

Wouldnt ln and e cancel eachother out though?

- anonymous

yes

- anonymous

So basically, to get k, you would divide ln(25/2) by 1.25?

- anonymous

yes

- anonymous

k=2.02?

- anonymous

yes

- anonymous

lol thanks. so to rewrite the equation, i would just plug everything in?

- anonymous

\[5000=400e ^{2.02(1.25)}\]

- anonymous

close but to be more general you use t instead of specific time

- anonymous

Ohh okay. So intead of 1.25, just t?

- anonymous

\[y=400e ^{2.02(t)}\]

- anonymous

that way you can find y for any t value you plug in

- anonymous

So on b, you would plug n as 5000 and y as 1500?

- anonymous

*15000

- anonymous

exactly , this time you are finding the 't'

- anonymous

I got 0.5438

- anonymous

Because you divide by 5000 on both sides.
Getting:
\[\ln(3)=(2.02)\ln(e)\]

- anonymous

And then divide by ln(2.02)
Did I do it right?

- anonymous

yes

- anonymous

Thank God. lol. But would that be like 5 mins?

- anonymous

what?

- anonymous

.5438

- anonymous

yes ,

- anonymous

it will take .5438 hours

- anonymous

okie dokes.

- anonymous

Thanks youuuu!!

- anonymous

nice to meet you

- anonymous

Nice to meet you too. It was good working with you.

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