A certain food is dropped on the floor and is growing bacteria exponentially according to the model y=2e^kt, where t is the time in seconds.
a) If there are 2 cells initially and 8 cells after 20 seconds, find the value of k for the bacteria.
b) The "5 second rule" says that if a person who drops food on the floor eats it within 5 seconds, there will be no harm. How much bacteria is on the food after 5 seconds?

Question

A certain food is dropped on the floor and is growing bacteria exponentially according to the model y=2e^kt, where t is the time in seconds.
a) If there are 2 cells initially and 8 cells after 20 seconds, find the value of k for the bacteria.
b) The "5 second rule" says that if a person who drops food on the floor eats it within 5 seconds, there will be no harm. How much bacteria is on the food after 5 seconds?

anonymous · Answer

You must do a) to be able to do b). Do you have any ideas for a)?

anonymous · Answer

Yes. I know it's $$8=2e ^{k(20)}$$ then divide each side by 2 and you're left with $$4=e ^{20k}$$ then you take the ln of both sides and get the ln4=$$lne ^{20k}$$ and then I got stuck.

anonymous · Answer

Wait I think I figured it out. Then you do ln4=20k and divide ln4 by 20 and get .0693147181

anonymous · Answer

Ok. Great start! 
$$4= 20e ^{k}$$ then $$\ln 4= ?$$

anonymous · Answer

yes. exactly!

anonymous · Answer

Okay so now for part b, do I substitute the k in for the k in the equation along with the 5 in the t spot?

anonymous · Answer

Yep!

anonymous · Answer

Okay so then I got $$y=2e ^{.3465735905}$$

anonymous · Answer

And divide by 2 and get y=2.828427125

anonymous · Answer

You don't have to divide anything by 2 there. You are just subbing in and the equation will give you a value for y. It will be 2 times whatever you get for the power of e.

anonymous · Answer

That's what I meant. That's the answer though correct?

anonymous · Answer

Since $$k = \frac{ \ln 4 }{ 20 }$$
You are taking e to the power of ln4 divided by 4, then multiplying that by 2.

anonymous · Answer

Can you just tell me if my answer is right?

anonymous · Answer

Just grabbing a calculator. One second.

anonymous · Answer

Okay thanks

anonymous · Answer

You are correct. Nice job.

anonymous · Answer

Thank you

anonymous · Answer

A certain bacteria is growing exponentially according to the model $$y=80e ^{kt}$$ where t is the time in minutes. a) If there are 80 cells initially and 675 cells after 30 minutes, find the value of k for the bacteria. 
I did that and got .0710895352 for k. Then I'm confused on b
b) When will the bacteria reach a population of 6000 cells?

anonymous · Answer

So you got k. This questions is similar to the last one, but also a bit different. What does the 6000 refer to?

anonymous · Answer

Never mind I figured it out myself!