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The answers to most of the following exercises are in terms of logarithms and exponentials. A calculator can be helpful, enabling you to express the answers in decimal form.A colony of bacteria is grown under ideal conditions in a laboratory so that the population increases exponentially with time. At the end of 3 hours there are 10,000 bacteria. At the end of 5 hours there are $40,000 .$ How many bacteria were present initially?

There should be $1,250$ bacteria in the colony to start.

Calculus 1 / AB

Calculus 2 / BC

Chapter 7

Integrals and Transcendental Functions

Section 2

Exponential Change and Separable Differential Equations

Functions

Trig Integrals

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you tell us about this colony of bacteria that they're growing, and they say that the population changes exponentially with time, and at the start of three hours there's 10,000 and after five hours there will be 4000 in this colony. And what we want to figure out is, what is our initial amount, right? So when we're told that we have exponential growth, it should look something like this for our solutions. So the population based off Time T is going to be equal to see times e to the K teeth. So let's plug in our two values here and actually instead of writing, see here. That's right, P don't. And this is essentially what we want to solve for what is P not because this is p of zero. Well, let's see. So let's first plug in p three and p five and see if we can get anything to help us with that. So we have p of three is equal to 10,000 which would be P not e to the K times three. Okay, and then where the other one would have p of by This is equal to 40,000 which is equal to e to the five K. Uh, P not one outside here. All right, so right now, it doesn't look like we do anything directly with this because we still don't know what p notice. But notice if we actually divided these two equations like this, then we're going to get 14 is equal to Well, the peanuts cancel out with each other, and then we're gonna have e to me three k minus five k, which should just be e to the negative. Okay. All right. Now, let's take the natural log on each side. And so it would cancel on the right with the e. And then we can divide each side by negative 1/2. So that would give us K is equal to negative 1/2 natural log of 14 fellas pull that negative 1/2 inside to be the power of 1/4. So that would first reciprocate it. And they would be four to the 1/2. Which is the same thing. Is square ruling it so that should give us K is equal to natural law. Go to. All right, let's go ahead and plug this into our original equation now, So we get P t is equal to peanuts. Yeah, raised to the K times natural log of two teeth times natural lava to cause Kay is natural too. And the nose we can use the power rule again to rewrite this as p not times natural law, go to t e and natural. Cancel out number is gonna look be left with p not times to tea for Petey right now. The reason why I want to do this is because this will be a little bit easier to plug in to solve for P NOC. At this point, we just use either of our initial conditions. I'm just gonna use p three to do this. Her p of three so p of three should equal to 10,000 be not times to raise to the third. And so that's going to be eight. So now we want to divide each side of our equation by eight, which tells us our initial population should be so 10,000 divided by eat. Looks like it will be 12 50. So this is the amount of bacteria that are calling. He should have started with

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