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A population of bacteria currently numbers 245. After 3 hours, the population is 290. Assuming exponential growth; after how many hours will the population reach 38...

Question

A population of bacteria currently numbers 245. After 3 hours, the population is 290. Assuming exponential growth; after how many hours will the population reach 3847

A population of bacteria currently numbers 245. After 3 hours, the population is 290. Assuming exponential growth; after how many hours will the population reach 3847



Answers

A bacteria population starts with 400 bacteria and grows at a rate of $ r(t) = (450.268)e^{1.12567t} $ bacteria per hour. How many bacteria will there be after three hours?

The problem. We have to get in a little tree. 4 50.268 years in power. On this 1.12 find 6 70 Do you do just given ass? 4 50 one. Do 68 years in general. Three, you devour 1.1 final 6 70 Did you get dizzy, Quinto? 4 50.2 68 That sort of this is to succeed in Do you have the power? 1.157 The one 111 Tau final 67 and zero in three, which is equal to 4 50.68 for well, one by one. Do find 67 Use about when. 11567 into to the minus. So this response to 400 review to power 1.1567 into Italy minus one which is even asked one want to We want to now the populace injuries by 101 23 hours. So the total population at three hours equals in the prison is 400 last 112312 religions to publicity invested in three hours to do do it will be 117 country bacteria. This is a big problem

Population starts with 400 bacteria and grows at a rate of R. F. T. Equals 4 50.268 E. to the 1.1256 70 bacteria per hour. How many bacteria will there be after three hours. Okay the number of bacteria Will be the initial population of 400 Plus. The integral from 0 to 3 For 50.268 Eat to the 1.1256 70 d. T. So we'll let you be 1.125670. So do you 1.12567 d. T. So 1.12567 needs to be in there. So one over that. Okay so we have 400 Plus 4 15.268 Over 1.12 567 E. To the U. D. U. If T zero U. Is zero If T. is three You is 1.12567 times three. 1.12 what? 1.12567 times three. 3.37701. Uh huh. Right so we get 400 plus let's go ahead and do that. Division. 4 50 .268 divided by 1.1 1.12 1.12567. So 400 Just about 400 times eat the you From 3.3770. 1- zero. So 400 plus 400 E. 23.37701 minus E. To zero. Yeah. Oops So 400 plus 400 times 3.37701. Eat that power minus one. I got 28.283 year times 400. Well plus 400. I get 11,713 bacteria.

We know that the rates are if he is simply the number of bacteria per unit at Time T used the equation. D B E T is equal to 4 50.26 82 The power for one times you can get D of T is equal to 450 0.268 point 268 e to the power off 1.12567 t divided by 1.12567 in order for self receiving plug in our initial condition, the coefficient should be equal to the initial population of 200. So we get C is equal to zero. So our final answer is falling for the population of easy. With three, we end up with 400 each of the power of 1.1225 times three, which is equal to 11713.23

Given that the rate of change of the population is proportional to the population at any given time that the initial population is 250 million. And that three hours later the population was 275 million. How long will it take for this population to reach? 300 million? We're going to be using this formula right here. And what we are missing. The particular piece of information that we need to find the time to 300 million is our is K. That growth constant. So let's start by finding that given this information that the initial population is 250 million. And three hours later it was 275 million. So starting there, let's take 275 million As are a. Which was three hours later, a note is 250 million. That was that initial population times E however, growth constant, which we're trying to find and our time was three hours later So we can divide both sides by 250 million which gives us 1.1 is equal to a to the power of K. Times three. Taking the natural log of both sides here so that we can get rid of that exponent on the right hand side allows us to get rid of the E. And we end up with Ellen of 1.1 times K. Times three. And Ellen are the natural log of 1.1 divided by three gives us K. Which is equal to zero 03177. This is a key piece of information that will need now And now to find the time to 300 million. We're going to do something very similar. But now the tea is the variable that we're missing. So moving on we now have 300 million is what we have on our left hand side Is equal to our initial population, which hasn't changed. It's still the 250 million that was given to us Times E. To the power of K. That we just found 0.03177 times T. Which is that variable again that we want to find. So let's divide both sides now by 250 million. Which gives us 1.2. Which is equal to then E. to the power of 0.03177 times teeth. Let's go ahead and take the natural log Now of both sides. Again that let's just pull this out of an ex opponent. Now dividing the natural log of 1.2 x 0.03177. We see that our time Is equal to 5.74 hours. So it would take about five and three quarter hours. So 5.74 hours. In order for this population to reach 300 million.


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