the-answers-to-most-of-the-following-exercises-are-in-terms-of-logarithms-and-exponentials-a-calc-11

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.
Biologists consider a species of animal or plant to be endangered if it is expected to become extinct within
20 years. If a certain species of wildlife is counted to have 1147 members at the present time, and the population has been steadily declining exponentially at an annual rate averaging $39 \%$ over the past 7 years, do you think the species is endangered? Explain your answer.

Bobby Barnes · Accepted Answer

So we&#x27;re told that something is considered in danger if after 20 years, the population is expected to be extinct or within 20 years. So we&#x27;re told that some wildlife has 1147 members at the current time, and population is following a steady decline exponentially, and it&#x27;s been averaging a rate of 39% over the past seven years. We want to determine if this species is going to be endangered and we&#x27;re gonna explain your answer. So here I write endangered if p of 20 is less than one. So if we have some population function pft Um, since they tell us it&#x27;s exponential decline, we know our equation should look something like this to start. So P of t is equal to p, not times e to the negative Katie, and this is because it&#x27;s declined. We have that negative there. And so we say, strictly less than one, as opposed to equal to zero, because exponential can ever equal to zero, at least with this model here. So that&#x27;s why we&#x27;re looking for less than one now. Let&#x27;s go ahead and figure out what to plug it. Soapy. Not is supposed to be our initial population. And we&#x27;re told initially it is 1147. So this is gonna be peanuts. So it&#x27;s got unplugged batted, and this is just from the equations that they give us in the chapter. So we don&#x27;t need to work harder than we need to, and we just plug everything in that they tell us we can use now. This next part, it says every year for the past seven years has been declining around 39%. So that means, after one year, soapy of one should be equal to zero point 61 of 11 47 which is p not so. We&#x27;ll just go ahead and set this equal to our function over here, and we&#x27;ll have e to the negative K, and we need to plug it one for teeth. Now. The reason why I wrote it out as six point one 4.61 as opposed to multiply and those together is notice that when we would divide by our initial population, those just can&#x27;t stop at this point, we would want to take the natural log on each side and then multiply by negative one and that&#x27;s going to give us. Kay is even too negative. Natural log of 0.61 So we have Kay. Now let&#x27;s go ahead and plug that into our original equation. So we&#x27;re gonna get that population after tea years is going to be a 1147 e to the So the negatives will counts out. So it should just be natural of 0.61 times teeth. Now we want to check to see if P of 20 is less than one. So let&#x27;s plug in 20. And actually, I&#x27;ll be a little bit lazy and we&#x27;ll just replace thes tease right here with 20. And so let&#x27;s see what we get after we plug. Everything is so this is going to be approximately 1107 and in the numerator 20 times. The natural log a point 61 is he to the negative 9.88 about and now exponentially ating. That gives us something around five times 10 to the negative fifth and then multiply that by 1147. So this is going to give us the population of about 0.5 So we did. We can&#x27;t have a fraction of an organism and it&#x27;s considered to be alive. So this is less than one which meets our criteria of being endangered. So we would just say something like, yes, since P of 20 is equal to 0.5 which is less than one.

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.

Bobby Barnes · Answer

So we&#x27;re told that this frozen human mummy was estimated to have died about 5000 years ago. And we want to determine how much of the original carbon 14 remained at the time of his discovery. Well, in example five, they tell us that for C 14 carbon dating, we will have our kay is equal to natural log of two over 5730. And the mount left will follow just the typical exponential decay equation. So for 5000 years ago, that means we&#x27;d want to plug 5000 in to that, so we&#x27;d have why 5000 is equal to. So why not times E to the negative K. So I was going to be natural log of two over 5000 730 and then we multiply this by 5000. So let&#x27;s see what that some natural law goes to times 5000 is about 3465. Then divide by negative 5000 730. So this is going to be why not times e to the negative zero 0.6 about and then exponentially eating. That should give about zero point by for six times. Why not? And so that we multiply it by 100 should be our percent left, which would give us about 54.6 per cent of those C 14 that would be left.

Bobby Barnes · Answer

So in 1940 we were told that this cave painting was found and it had about 15% of the original carbon 14 remaining. We want to estimate how old is the painting? So let&#x27;s go ahead and see. So an example. Five. They tell us for carbon 14 dating that are K is going to be natural to over 5730 of reciprocal years and this fall&#x27;s exponential decay since it&#x27;s radioactive. So if we want to find how long it took for 15 years or 15% to be left, we would just plug in 0.15 Why not? Which is our initial amount on the left side, and then we&#x27;d want to solve or K. So I&#x27;m just gonna for tea. So I&#x27;m just gonna leave Kay, though as K for right now. And we can plug that into the variant. So first I&#x27;m going to do divide your side by. Why not get those cancel, take natural log and divide by negative k. So doing all that would give T is even to the negative metro log of 0.15 over. Okay, Now we can plug in what K is to get negative 5730 natural log of 0.15 over the natural log of two, and this would be in years. So let&#x27;s go ahead and see what this is approximately so negative. 5730 times natural like a 0.15 is about 10,000 or about 11,000. Then divide that by natural law goes to and that is going to give 15,682. So this is how old it was in 1940 so years mold in 1940. And then, if you want to be more specific, you can just go ahead and ad how many ever years it currently is to this to get its true H.

Bobby Barnes · Answer

so we have that. U. S Census Bureau has a running clock with US population. So on September 20th 2012 the rate was increasing at one person for every 12 seconds and the population figure at 8:11 p.m. Eastern standard time on that day. WAAS 314,490,198 people. We want to assume exponential growth at a constant rate and then find the rate at which the populations or find the great constant or the population. So since they tell us to assume exponential group, that means we can write this in the following way. So it&#x27;s going to be why is equal to or maybe all right, p a t for population. So P A T is equal to p, not times e to the k t. And we want to solve for what cases. So they tell us that the population is increasing at one person for every 12 seconds, so just so the numbers don&#x27;t get too unruly, I think it would be a good idea to comfort that too people per day. So let&#x27;s see, so one person per 12 seconds so Let&#x27;s think about this. So there is 60 seconds in a minute, so it be 60 seconds her minute and then there&#x27;s 60 minutes in an hour. And then lastly, we go ahead of this line over a bet. There are 24 hours in a day, So this is going to give us a rate. Oh, so let&#x27;s see one. So we have 60 times, 60 times 24 then we divide that by 12. So there&#x27;s gonna be about 7200 people her day. So we have this here, so after one day, we should have. So right now we&#x27;re using time in days, soapy one is going to be equal to So Pena is our initial population. So 31 or actually, before we do that, we need to add how many people we gained would be 72,000 plus our initial population, which is that number. We have a top there, and then it&#x27;s going to be equal to 3141 or 91 by eight, and then we have e to the just k. So let&#x27;s go ahead and add the used to see what we get. So it looks like we&#x27;ll get 31 or or to six re 98 once we add those, and then we would want to divide by this the 31 or or 19198 Then we would want to take the natural log on each side. So I&#x27;m just gonna go ahead and do that so we don&#x27;t need to do a bunch of algebra for anything. So that&#x27;s going to give us that K is equal to this value right here, which is approximately equal to, let&#x27;s see. Well, it looks like it&#x27;s around 2.2 a times 10 to the negative fit. But it&#x27;s important what our units work for this. So this is going to be and her day. But we wanted it in her years up here, so we would just need to convert this by multiplying by. Well, there&#x27;s 365 days in every one year, and so that&#x27;s going to give us approximately equal to zero point 008 three bye eight. So this would be her year for people her year or our great constant. All right. Actually, I guess it&#x27;s really just per year and if we want to write it out. Exactly. We would just multiply what we got over here by 3 65 No, that&#x27;s right. That don&#x27;t well for the speaker number. All right, so this would be our solution or K Now, for the next one, they want us to figure out what is our population after seven years. All right, well, that means we want to take our original equation and plug in seven years. So when we worked with it in this form here, we were using it. Or actually, you don&#x27;t even need to worry about there because our unit over here is in years now. So we could just plug in seven years in prick a, and we&#x27;ll have no issues. So we have 314 or 19198 e toothy. So seven times our value for K, which is a round zero points. 0083 Buy paint, which is around the value of Let&#x27;s see so e to that number. Looks like it&#x27;s around one point of six that we multiply it by our initial population. So we&#x27;d have 33336378 Bo, let&#x27;s put some commas in here, so it looks like our population, after seven years will be about 333,362,780 people.

Bobby Barnes · Answer

So they tell us. In Whittier, Calif. Warning of the amount of oil that is pumped from one of the canyon wolves decreases that a continuous rate 10% for you. We want to determine when will the Wells output Balto 1/5 of its present value. All right, so this sounds like an exponential because it&#x27;s saying it&#x27;s decreasing by 10% every single year. So we would write out that. So if Wyatt Tee is supposed to be how much oil we are pumping, there&#x27;s gonna be Y t is equal. Why not e to the K? T? All right. And we want to first solve for Kay. And so they tell us it decreases by 10% her year. So we know why of one is going to be will decrease by 10%. Should be 0.9 of our original amount like this. And then we would divide by white, not meet side so they cancel out, and then we would take the natural log of 0.9, and I almost forgot to plug in Ah t here, who&#x27;s this? Should have been one. And so then taking the natural log on each side gives us K is equal to the natural log of zero point. Not all right now they want us to solve for win is going to be one of us. Let&#x27;s plug K in First was gonna be Why not? Times e to the t natural log of zero point signed. And we want this to be equal to one here. Why not? So again, the why nots we&#x27;re just gonna cancel each other and we would take natural log on each side to get the natural log of one is equal to t natural off of zero point. Now, we would want to go ahead and divide each side by natural light of 0.9. So that&#x27;s gonna give us t is equal to the natural log of 1/5 which is 0.2 over natural log of 0.9. And let&#x27;s see, this is going to be approximately natural. Aga 0.2 is about 1.6 and then divided that by natural aga 0.9 this gives about 15.3 years. So after about this time, well, we have got to 1/5 of be production

Problem

The answers to most of the following exercises ar…

Problem 33 Hard Difficulty

Answer

Yes, $y(20)<1$

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Yes, $y(20)<1$

University Calculus: Early Transcendentals

Catherine R.

Anna Marie V.

Kristen K.

Samuel H.

Integration Review - Intro

Definite vs Indefinite

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

Catherine R.

Anna Marie V.

Kristen K.

Samuel H.

Joel Hass, Christopher Heil, Przemyslaw Bogacki

University Calculus: Early Transcendentals