The organism E. coli is a common bacterium. Under certain...

The organism E. coli is a common bacterium. Under certain conditions, it undergoes cell division approximately each 20 minutes. During cell division, each cell divides into cells. (a) Explain why the number of E. Coli cells present is an exponential function of time. The number of E. Coli present is an exponential function because the number of cells is doubling each 20 minutes, representing constant proportional change (b) What is the hourly growth factor for E. coli? 8 (c) Express the population N of E. coli as an exponential function of time t measured in hours. (Use $N_0$ to denote the initial population, and t to denote the time in hours.) $N = N_0 \cdot 8^t$ (d) How long will it take a population of E. coli to quintuple in size? (Round your answer to two decimal places.) hr

Transcript

00:02 Okay, so in this problem, we're told that a e.

00:07 Coli cell divides every 20 minutes and letting y of t equal to the number of cells at time t.

00:22 We are told to work out some things about this y of t, assuming that it's an exponential model, as in y of t as in y of t as a exponential function.

00:34 Let's first sort of draw the problem to get a kind of handle on what's going on.

00:40 So we start with a single cell and imagine sort of time axis going down this way.

00:46 And after 20 minutes, it's divided into two.

00:51 And these cells are identical to this cell, right? i mean, you can kind of assume that.

00:57 And so you would assume that after another 20 minutes, after 40 minutes in total, these cells have each divided.

01:06 And so on.

01:09 After 60 minutes, get more cells, so on.

01:14 And so we can kind of already see just from this picture.

01:18 This is a binary tree, right? and so we know that it's going to be an exponential model because you're just doubling every time, right? so times two cells from 1 to 2, times 2 from 2 to 1, 2, 3, 4, and then to a, you multiply by two.

01:38 So we know it's going to look something like 2 to the n for some kind of n.

01:44 Obviously, this is a discrete description of it, and we want a continuous description, but this kind of gives us an idea of what the model is going to look like.

01:54 So moving on to the actual parts of the question, we have first part a, which just asks us to set up an initial value problem whose solution is y of t.

02:06 So what information do we have? we have that first that y of t is exponential.

02:15 And so if we kind of remember what an exponential model looks like, a general exponential model is going to simply be something like this, right? this kind of encapsulates every possible exponential model that could be.

02:31 You just have to change a and b.

02:32 And of course, a and b here in this case, i'll probably go.

02:37 Going to be real numbers.

02:40 I mean, there's cases.

02:40 They could be complex numbers, but i don't think they would.

02:44 So anyway, yeah.

02:46 And we have, what information do we have? well, it's an initial value problem.

02:51 So what's our sort of start and end values that we know about y of t? well, it's told us that we start with a single molecule or cell.

03:03 And so we know that y zero is one.

03:06 And it says that after 20 minutes, the cell has divided into two cells.

03:11 So y of 20, choosing our time units to be minutes here, just to make it a bit easier, we know that y of 20 is two.

03:23 So there we go.

03:23 That's our initial value problem.

03:26 That's what it requires, really.

03:28 That's set up and it can be solved from there.

03:31 And in fact, in part b, it asks us to solve the initial value problem to find a formula for yft.

03:40 So let's do that.

03:42 Well, we know y of 0 is 1.

03:47 And so let's find a first.

03:50 We know t is 0 in this case.

03:52 So plugging it in, we have a times e to the b times 0.

03:58 Well, b times 0 is 0 and e to the 0 is 1...

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