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Colony of bacteria is grown under ideal conditions in a laboratory so that the population increases exponentially with time. At the end of 2 hours there are 40,000 ...

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Colony of bacteria is grown under ideal conditions in a laboratory so that the population increases exponentially with time. At the end of 2 hours there are 40,000 bacteria: At the end of 6 hours there are 50,000. How many bacteria were initially present?There were bacteria present initially: (Round to the nearest whole number as needed:)

colony of bacteria is grown under ideal conditions in a laboratory so that the population increases exponentially with time. At the end of 2 hours there are 40,000 bacteria: At the end of 6 hours there are 50,000. How many bacteria were initially present? There were bacteria present initially: (Round to the nearest whole number as needed:)



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Growth of bacteria 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?

Section 64 Problem 24. We're dealing with the growth of bacteria. So we've got bacteria that can be modeled with exponential growth and were told after three hours, we have 10,000 bacteria. After five hours, we have 40,000. How many were president initially? So we're governed by some model here. Why is equal to why subzero and then e to the Katie. Let me write that a little bit better. So e to the Katie. So this is our exponential model. Now what we know at this point we know that 10,000 So 10,000 is equal to Weiss of zero e to the three. Okay. And we know that 40,000 is equal to wise of zero e for the five k. So if we take the ratio of those, then I know that 40,000 divided by 10,000. It's going to be equal to y subzero e to the five k over Why subzero e to the three k And then what this gives me is on the left side 40,000 over 10,000. You have four is equal to e to the two k. So if I take the natural log of both signs. Natural log of four is equal to two K. Therefore, K is equal to the natural log of four divided by two. And this turns out to be about point 69 31 Okay, so that gives us the value of K in this exponential model. So now we're down to why is equal to why some zero e and then it's raised to the point 6931 So 69310.69 31 t. So this is my exponential model. Okay, now, I knew one of the values that was had what? At three hours, I get 10,000. So I know that 10,000 is equal to wise of zero e to the 00.6931 times three. So why subzero is equal to 10,000 divided by E to the three time 0.69 31 OK, and then all of this goes into a calculator and I come up with 12. 50 0.18 If we go back to the original problem. We're talking about the number of bacteria in growth. Okay, So doesn't make sense to say 12 50.18 Bacteria's are really better answer is about 1250 bacteria initially present. So what this would mean is my model is going to be wise equal to 12. 50 e to the 0.69 three won t so that would be the exponential model that

But the initial population and we're trying to find the double time. So, um, we are told that at 10 hours the bacteria was at 5000 and then at 12 hours were told that the bacteria was at 6000 and that it's proportional. So we're gonna set up in proportion. We're gonna dio 5000 over 6000 is equal tissue population times E to the 10-K powers. So I was 10 for that top part, and then the initial population times E to the 12 k for 12 hours at the 6000. So the 5000 over six thousands going to reduce to five over X. The initial populations are gonna cancel out for now and weaken Divide eat of the 10-K over each of the 12 K by subtracting the exponents. So 10 minus 12 is negative two. So we're gonna have e to the negative too. Okay, so we're gonna raise both sides to the natural log in order get the e and the natural after cancel, we're gonna divide both sides. I negative to, And what this accomplishes is that gives us our K, which we need in order to find the initial population and the double time. So Natural Lago 56 media close up print senior calculator. Get that answer and divide that by negative too. And you should get a proper points. You're not grounded to do. It's We're gonna take that 0.9 now and we're going to plug it back into one of the equation so you could do 5000 equals p the initial population times E to 10-K or 10 times 100.9 or you do 6000 equals. Finish the population times E to the 12 times point. Oh, night, it doesn't matter. So I'm going to go 5000. So 5000 is equal to the population Trying to find times e raised to the 50000.0, nine times Ted power. And so you're gonna divide both sides by point by eat of the porno nine times 10 and you're gonna want to close that in Brent Theses 0.9 times 10. And if you do that correctly, you should get up initial population of 2032 Now that's not the last thing we need to find. We need to find out the d t the little time. So double time is one divided by 10.9 times the natural log to from art section in the book. And we do want about my 0.9 times natural look of two, you're going approximately 7.7 hours.

A bacterial colony grown in a lab is not to double in number in 12 hours. Suppose initially there are 1000 bacteria present in part A. We use the exponential function Q equals Q zero times E. To the K. Times T. To determine the value K. Which is the growth rate of the bacteria. We round the result for decimal places. And in part three we determine approximately how long it takes for to 200,000 bacteria to grow. So there um The first thing to notice is that 40 equals zero. We have 1000 bacteria present. Yeah it means that Q. At zero which is equal to Q zero. E. To the K. Time zero. That is Q zero because each of the zero is one Is equal to 1000 Because it's cute at time zero. So 00. Sorry this year is 1000. I meant that's what I found here. And it means that the function Q. Is equal to 1000 E. To the gay times T. So that's our first result. Now we know that the bacteria double in number in 12 hours. It means that if we evaluate the function at time 12 and I know that the time is measured here in hours. It's very important. So if we violate the function of 10 12 We will have the double of the number of bacteria we had initially. That is the double of 1000. That is 2000. So the number of bacteria at a time T. 12 must be two times 1000. That is 2000 because it doubles it's number after 12 hours. So q. of 12. You at 10. 12 equals 2000. And we now we can use the formula of Q. Which we have here at the time being so with 1000 E. Okay times 12 Because we are evaluating at T equals 12 And that had to be equal 2000. We pass 2000 to the left. Sorry 1000 to the right. Better we get that E. To the 12 K equals two. 2000 over 1000 equal to. And so we take natural algorithms to find the exponent that is 12K is equal to the natural logarithms of two. That's because the natural algorithmic see inverse function of the exponential function And then K. equals natural rhythm of to over 12. And if we calculate this value Isn't a calculator and round into four days emotes We get to 0.0 578. Maybe because we have a zero after the point it would be maybe better to run to five or six decimal. But we stick to the statement and yet before designates and having this result here we can say that The exponential function q. of tea is then 1000 times E. To the 0.057. eight times G. So we have completely determined the function Q. Using the two informations about the initial number of bacteria present and the behavior of the number of courts bacteria. That is that it doubles after 12 hours. So this is bar a non pervy. We are going to calculate or estimate how long it takes for 200 200,000 bacteria to grow. And that means that we want to find time correspondent to that number of material. That is The equation we gotta solve is 200,000 equal kft equal 1000 E. to the 0.0578 T. We got to find time. So again We do that calculation here by passing this 1000 divide into the left. And we get E. To the 0.0578 T. Is equal to 200. And so taking natural liberty in both eyes we get 0.0578 T. Is equal to the natural logarithms of 200. And then he is The natural algorithm of 200 over 0.0578. And if we use a calculator disease Approximately equal to 91 point 666. So uh this is even in hours. So to have a better idea. We divided by 24 We had that goes to three. And then the reminder to To get the number 91.666 is about 20 hours. So it's almost four days. So the time required 40 bacteria for um he's put a number of very trust four. 200,000 bacteria to grow every time I meant For 200,000 material to girl. It's about three days because 24 10 3 is 72 hours. It's not four because times four is 96 greater than that. So in three days and the remainder Which is 91.666 -2014 3. Yes. Any almost 20. So with three days and 20 hours At the time required for bacteria to growth to the number 200,000 Or equivalently to have in the colony into lab to have 200,000 materials. And the information comes from the fact we are using an exponential function of the form to zero times exponential of Katie with the growth rate of K. And that's why the case is determined by knowing the behavior of the bacteria. That is that it doubles The number in 12 hours. And that initially we had 1000 material with all that information we determine completely the function Q. The rate including the rate of growth of the bacteria. And we found the time require For the bacteria to grow to a number of 200,000

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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