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Suppose that the rate of change at any time $t,$ of the number of bacteria is proportional to the population at any instant. Furthermore, suppose, initially, the population is 250 million and three hours later the population is 275 million. How long does it take for the population to become 300 million?

5.74 hours

Algebra

Chapter 4

Exponential and Logarithmic Functions

Section 7

Applications of Exponential and Logarithmic Functions

Missouri State University

McMaster University

Lectures

02:00

The rate of change at any …

02:27

A population of bacteria $…

03:23

Population Growth A popula…

03:43

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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