the-answers-to-most-of-the-following-exercises-are-in-terms-of-logarithms-and-exponentials-a-calcu-8

Question

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?

Bobby Barnes · Accepted Answer

you tell us about this colony of bacteria that they&#x27;re growing, and they say that the population changes exponentially with time, and at the start of three hours there&#x27;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&#x27;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&#x27;s plug in our two values here and actually instead of writing, see here. That&#x27;s right, P don&#x27;t. And this is essentially what we want to solve for what is P not because this is p of zero. Well, let&#x27;s see. So let&#x27;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&#x27;t look like we do anything directly with this because we still don&#x27;t know what p notice. But notice if we actually divided these two equations like this, then we&#x27;re going to get 14 is equal to Well, the peanuts cancel out with each other, and then we&#x27;re gonna have e to me three k minus five k, which should just be e to the negative. Okay. All right. Now, let&#x27;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&#x27;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&#x27;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&#x27;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

Bobby Barnes · Answer

they tell us about how this colony of bacteria can start with one bacterium and doubles every half hour. Ed, it has an exponential growth, or we want to determine by the end of 24 hours How many bacteria will this colony have? So what&#x27;s seat? So since they tell us his exponential growth, we can already know that we&#x27;re going to set this up to be so I&#x27;m using P to be the population of our quality. So there&#x27;s gonna be P A. T is equal to see Time&#x27;s E to the K T. All right, so we have this equation. Well, let&#x27;s go ahead and use our initial condition here to solve for C. So that&#x27;s p of zero is a good one. You got a C E o to the k t eat k zero, which that&#x27;s just gonna b E to zero, which becomes not 01 So then that tells us that see, you should just be warned so we can go ahead. And right now, Petey is just equal to e t k t. Now we need to solve for Kay, and we can use the fact that the population should double every half hour. So if our population started at one that&#x27;s gonna tell us that p of 1/2 should be equal to to get us just double. Forget about two is even too e to the 1/2 K. Now we will take the natural law on each side, and that would cancel out and we&#x27;d have 1/2 K so it multiplied by to get Kay is secretive, two natural to which we can rewrite as the natural log of or using the X golden rule for lager rhythms. All right, so let&#x27;s write out what P a. T is, and then we can just plug in 24. So this is going to be he two. The tee times natural log. This should be e to the natural log. Oh, tee times the natural log for I don&#x27;t know why I was I have a hard time figuring out what You&#x27;re right up there. Now we want to go ahead and plug in 24 but actually first knows that weaken, right? This actually in the following way, we could pool in that tea to get natural aga or to the T and in the E in the natural law caps out with each other and we can write. This is just for to the teeth. So this is gonna be P A t. Now let&#x27;s plug in the fact that we want to check for 24 which would give us or raise to the 24 on Let&#x27;s see what the best numbers or raised to the 24. Also, there&#x27;s a huge number, so it&#x27;s approximately equal to two times two times points or 2.1 eight times 10 to the 14 back till you. So if you want the exact answer, it&#x27;s four to the 24 approximately that 2.81 times 10 to the 14.

Brittany Knowlton · Answer

so his father look at the growth of bacteria were told a cognac too is grown in ideal conditions. Love tourists at the published increases exponentially with time so increasing exponentially. So I know for all of these that we&#x27;re just gonna be having equation. Why is equal Tio? Why not? Hey, to the Katie. So next, looking at that next sentence, it says at the end of three hours, there are 10,000 bacteria. So this why here is telling us how much bacteria we have at the end of this certain amount time. So were first, given that after three hours we have 10,000 bacteria and the nexus is at the end of five hours. There are 40,000 bacteria we have wide of five this equal tio 40,000 and were asked how many bacteria that were initially. So we&#x27;re looking for the value of why not? So I was when the plug and stuff that I know it first. So that&#x27;s why a three&#x27;s equaled in 10,000 that is telling me that 10,000 people, too. Why not e to the three for tea? So three, Okay. And similarly, this one is telling me that 40,000 Siegel to Why not? E to the five came. So in order to find this, why not? I do need to find my k first so that I can use one of these equations to find my Why not? So I want to start these equal to each other so I can cancel. Ease out. I&#x27;m really close to it. To make this one equal to that one. I just need to multiply both sides of this equation by four. Take my 40,000. And so now this is equal to his bottom. So I get for you. Why not? It&#x27;s a three k is not equal, Tio. Why not? Yeah, since five K. And so now I&#x27;m just going to solve. That is normal. I see that my wine not to cancel out. So I&#x27;m just left with for each of the three k this equal to eat the five k next one. Divide this Each of three cam. Both sides, like it for is equal to e to the two k. And so then solve enough. I taken natural log of both sides. I get natural law. Before she was the natural log of EADS that&#x27;s UK, this national before. Same thing is to not your log to using the power rule. So therefore I get that Kay is equal to the natural log of team. So now that I have all of my variables in here, I can plug all those in to find the value of why not. So I want to go ahead and use this first equation and plugging all my values. So I have that 10,000 sequel to Why Not? E to the K, which we found was natural over two times. Three the time using the power roll this three times. And I thought to use the same thing as the national organ Too cute, which is National of eight, signifying further eats natural, eh, the same thing as eight. So therefore dividing eight on both sides. I get that the initial amount of bacteria was 1,200 and 50

Robert Daugherty · Answer

Section 64 Problem 24. We&#x27;re dealing with the growth of bacteria. So we&#x27;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&#x27;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&#x27;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&#x27;re down to why is equal to why some zero e and then it&#x27;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&#x27;re talking about the number of bacteria in growth. Okay, So doesn&#x27;t make sense to say 12 50.18 Bacteria&#x27;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

Manan Sheel · Answer

hi and this Christian related to bacteria Groot bacteria culture are the is eight are the change in the population of bacteria or time to multiply their population. So it is said that there&#x27;s a certain culture of bacteria is called Steptoe focus. And initially we have done step to bocas bacteria and in every 1.5 hours double like for the 1st 1.5 are they become 20. After three hours you come 40 writers. Okay, so So we&#x27;re asked asked the discretion is three parts in the first part is it is asked that we have to find an exponential model of this bacteria they the number of bacteria after time t it could and not times toe t by 80 to find this modern based on aware vision of statement Soviet This and not here is the Narcan, the formula that just had noted the initial population of the bacteria in this case it is then and this a Here is the time after which after it the victory becomes double. So in our case is 1.5 hours and initial population of bacteria is stan and in this from the unit softy and air seemed genuine that he should be in hours or days. When friends, it&#x27;s fine. So the model equation becomes an anti is equal to and not is 10 times Tour de Wide way one point fire. Okay, this is the solution of the first part. It sounds so weird in AS and D. We could be 10 times if you expand 1.5 and comes to your do 10 dance to deal with three to using this family in the next part. If you complete the number off, the more arbitrary after 35 hours of his disease. Ted, if I would be the number off bacteria and 35 there&#x27;s 10 times two to the power to 35 were three. Did you goto 10 times two to the power 70 or three. Okay, let&#x27;s find What is this scapular? This good sten times to do our 70 or three, and it&#x27;s that comes out to be around by under six million bacteria. So it is around 10 fine. 689 838 Stevia with when it&#x27;s all rendered five million bacteria and the sea parted is asked, is in the same model, illegal plate is given that created time, after which the bacteria count becomes or the anti becomes 10,000. So in this, the fund will let it be a derived. This model, if you put anti, is equal to 10,000. Good with the time it is 10,000 visible to 10 times two to the power duty or three. And this is 1000 record do to really power two d or three. They&#x27;re staking out to the base 10. On both sides, Miss becomes long to the based 10 north tenderly our tree to go into low to the based 10 off. You went to the room to go three the left hand side of just three going to cooperative the law in this history and Megan rented says to to a tree and put in front to the property Off log and love to the based standoff Do so. This becomes two D is nine hours to times Rio goes to Okay, so this is our be Let&#x27;s get rid this rain new hiding by two way. So we&#x27;re down 15 hours 14.94 hours to be took for the publishing of bacteria produced about 10,000 funding by 94 hours. Whatever solution to part. See? And this one of a solution to part B mochi. Thank your falling.

Bobby Barnes · Answer

So we want to solve this differential equation that they give us here that talks about how much is left in a chemical reaction. And we&#x27;re told initially there&#x27;s 100 grams and then we want to figure out how many how much is left after one hour. So this looks like a separable differential Equations. Let&#x27;s first divide each side by Why? So this is going to give us won over Why? And I&#x27;m going to multiply the DT overwhelm attitude so we should have d y is even too negative. 0.6 80. Now let&#x27;s integrate each side. So on the left, we know that&#x27;s going to be the natural log of the absolute value of why and on the right, that should be negative. 0.6 t plus some constantly called C one. No, it doesn&#x27;t matter for the absent value in this case, because if you think about it, why is telling us a mass that we have left over and it doesn&#x27;t make sense to talk about negative mass is so we don&#x27;t need the opposite value, so we could just exponentially in each side it was canceled, and we&#x27;re gonna be left with Why is equal to E to the negative 0.60 plus C one. Now remember what we have powers being added something being added in an experiment. I should say we could write it as the multiplication of two expressions like this, and then e to the C one of just another constant. Let&#x27;s just call that see now. So this is going to be our equation, or at least our general solution of this differential equation. And now let&#x27;s go ahead and use our initial condition that why zero should be equal to 100. And remember, the way this is red is that T is equal to zero. So we&#x27;re gonna go ahead and plug zero into. That&#x27;s going to be C Times E raise to the world. Negative 016 times zero is zero, and then that&#x27;s just going to be one. So that tells a C is 100. So now we&#x27;re going to have why of tea is equal to 100 e to the negative 0.6 t. And if we want to figure out for one hour later, which plug in one why of one is equal to 100 times e to the negative 0.6 times one, which just leave us with that. And now let&#x27;s check out what this is approximately equal to so e to the negative 0.6 about 0.54 Multiply that by 100 and this should give us something around 54.8 grams. So after one hour, this is how much of our reactant will be left.

Problem

The answers to most of the following exercises ar…

Problem 30 Hard Difficulty

Answer

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

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

University Calculus: Early Transcendentals

Catherine R.

Kristen K.

Samuel H.

Joseph L.

Integration Review - Intro

Definite vs Indefinite

Joel Hass, Christopher Heil, Przemyslaw BogackiUniversity Calculus: Early Transcendentals

Catherine R.

Kristen K.

Samuel H.

Joseph L.

Joel Hass, Christopher Heil, Przemyslaw Bogacki

University Calculus: Early Transcendentals