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(15 pts: )Write down the differential equation for modeling population under the assumption that the population grows with constant relative erowth rate Show how to...

Question

(15 pts: )Write down the differential equation for modeling population under the assumption that the population grows with constant relative erowth rate Show how to derive the solution P= Aekt . of this differential equation:(iii) If we uSE this model for population of country for which the population in the year 2000 30 million , and the population in the year 2010 was 35 million what would the model predict the population would be in the year 21502 Suppose instead the logistic differential equ

(15 pts: ) Write down the differential equation for modeling population under the assumption that the population grows with constant relative erowth rate Show how to derive the solution P= Aekt . of this differential equation: (iii) If we uSE this model for population of country for which the population in the year 2000 30 million , and the population in the year 2010 was 35 million what would the model predict the population would be in the year 21502 Suppose instead the logistic differential equation to model this country $ population where use the same value for k as 4Ou determined above and wee use the value of M 100 million. (iv) Write down the logistic differential equation (do not attempt meaning of the constant M_ solve it) and explain the Recall that the solution of this differential equation can be written as M _ Po where A P(t) = 1 + Ae-kt Determine the value that this model predicts for the population in the Fear 2150. (vi) According this model when will the population reach 50 million?



Answers

In this section, several models are presented and the solution of the associated differential equation is given. Later in the chapter, we present methods for solving these differential equations. Logistic population growth Widely used models for population growth involve the logistic equation $P^{\prime}(t)=r P\left(1-\frac{P}{K}\right)$ where $P(t)$ is the population, for $t \geq 0,$ and $r>0$ and $K>0$ are given constants. a. Verify by substitution that the general solution of the equation is $P(t)=\frac{K}{1+C e^{-r t}},$ where $C$ is an arbitrary constant. b. Find the value of $C$ that corresponds to the initial condition $P(0)=50$ c. Graph the solution for $P(0)=50, r=0.1,$ and $K=300$ d. Find $\lim _{t \rightarrow \infty} P(t)$ and check that the result is consistent with the graph in part (c).

All right, so let's have a look at Question nineteen from Sexual or Chapter nine Raid's. Let's modify the logistic, the French for equation of example. One. This falls. So we are to consider this equation. DP over dt because Point I p Times one minus p over thousand minus fifty. The first question reads. So let's suppose it pfft represents a fish population of time. Tea with he's married Weeks explained the meaning ofthe the final term of the equation negative. So this leg of the fifteen here represents a decrease or fifteen fish for a unit of time, which is weeks that's constant throughout the evolution. So one way to interpret this is to think that, say, there's a fisherman who will care fifteen fish every week, no matter what, no matter how big or small. The population of fish in the lake, let's say is you always catch fifteen fish every week. That's why I have a constant night of fifteen. Turn here in the equation for the derivative after fish population proof, very time. So it's a simple simple is that kind of a Now I don't b s is to draw a direction field for this equation. Now, before we do that, it's a good idea to go ahead and factor the left hand side of this threats by the right hand side of disagree, it means she'll express it as a product of two terms. Ah, because that will make the whole question a lot easier. So how do we do there? Well, it could. He should start by distributing this product. Right? Because if you do that, we're gonna end up with a quadratic equation. All right? It's gonna look like it don't look like, uh, point Oh, a B minus point will be spread over a thousand and fifty, and so that's a quadratic equation. Can use the quadratic formula to factor it. So I'm going to go ahead on DH, skip that arithmetic and just tell you if you use the quadratic formula to find the roots, you will find the roots to one hundred and fifty, seven hundred and fifty. So if your Route two hundred fifty and seven hundred fifty factoring looks like this and you have a coefficient here at the beginning which is the coefficient ofthe B squared in your equation, which in our case is negative Final eight divided by a thousand. So that will give us names. You're a zero zero zero rate. That's our equation factor. Did just He's the quadratic formula to find these roots. And then at the coefficient in the front. So now that we have factored our equation, it becomes a whole lot simpler to draw. Ah, direction Field is Ah, let's add a new when they're here. So let's imagine a graph. Sorry. Imagine the Griff here, where this axis represents the time Gladys and weeks and this backs is represents the fish population. Right. So we have, you know, that at two hundred fifty and that at seven hundred and fifty this slow zero, that's what the equation we have previously thousands. So at both two hundred and fifty and seven hundred fifty, I can just draw as zero slow controlled this more accurately. All right? He has your slope. And also for seven. Fifty, you have zero slope. Something like that. Now, what about elsewhere? Well, noticed that this is a quadratic equation with a negative coefficient. Right? So what? The quadratic equations of negative coefficients looks like they looked like downward pointing prevalence. So between the two routes. You have something positive. And outside of both fruits, you have something negative. So, Francis here, When you go up from seven. Fifty, you're going outside of the roots. So you're going to start to have more and more negative slopes. So you're slow starts around here. Negative, but not very large. Say, and then it starts to become more and more native. The Freddy you go on any start negative, Not very large. And it becomes on negative. They're not very large. And you get more than one something like this. That's how parappa this work. And the same thing works here under two. Fifty. Or you're also going away from the root. So you start over negative slope. That's not very large. And that's so gross is you go through yours. No, it doesn't. So let's go look something like that. Now, here between two hundred fifteen seven hundred fifty, you're gonna have positive slopes, right? So say, starting here to fifty when you go a bit above, you gotta have a positive slope that's also not very large. Try to draw this less light, and then your slopes will begin to grow. Right. But then when you get closer to seven fifty day with decrees again, because they have to be zero when you get your seven fifty. So it works something like that, right? You have slopes like grow a little bit, then a decrease back down where they're always pasta. Here, write something like this at So it's not a very good drawing, but help pick. Explanation Mason. So this is some what what the but the direction field would look like, remembering that at seven fifteen to fifty, you have zero smoke. All right, so for birth sea Where are the equalization solutions? So equilibrium is exactly when the fish populations in equilibrium, which means it doesn't change. So that's what we've just figured out. It's seven fifty to fifty. That's when you get a constant line for the amount of fish. So if you start with either seven fifty or two fifty fish, then you'LL always have that state. Now it'LL never change those air. The equilibrium solution. Then party ass is. She used the direction few to sketch several solution curves, so let's have a look at what happens. So supposing start off a population of fish that's less than two fifty say that's supposed to start here now. The slopes are negative cigarettes that you go down, and in fact, as you go down the slopes get more and more negative. So you going to go down quicker and quicker and quicker, quick, and you're going to reach zero in a finer stock. Since you're your fish population is decreasing at a steadily increasingly negative wrecked it. So any fish population below two fifty is going to decrease rapidly and reach zero in a final time. Now what happens if you start off a fish population that's the above two fifty? Well, here the slopes are increasing, so you're going to increase. But as you get closer to seven fifty, your population starts. The smoke. Such decrease and say's you're increase becomes slower and slower and slower and slower. In fact, they will have seven fifty as a horizontal acid, so your fish population will steadily increase, but at a slower and slower rate and will attend at infinity towards seven fifty without actually ever reaching seven fifty. So that's what happens if it's top of the first population that's above two fifty, but smaller than seven fifty now, what happens if you start off a fish population that's greater than seven. Fifty? Well, saving say, start here or your slopes are negative. So it's going to you start to decrease, and it will keep increasing, but your slopes again in closing closing zero. So you're decrease will get slower and slower and slower again. You have a horizontal ass until at seven. Fifty, but you're never actually going to reach fifty. So if you start with a population of about fifty, it decreases, but at a slower and slower rate. And it tends towards some fifty and finish without ever actually screeching seven fifteen, at least in this small off course. So that's a heuristic sketch off what the solution curves looking like look like, according to our drawing off, the director of Shield. All right. So finally, Bar E asked us to solve this the French equation Explicitly. This one here, this he could have this. All right, so how do we solve this explicitly? Well, this is a separate ble equation, so we can write it thus just right. We can put our peace on one side of the equation off together, off the DP and we can put everything else of the other side together with the DDT, So it'LL look something like this. Over here, we have B and will be provided by me. Linus, you're fifty seven. Fifty. Right? And over there we have this proficient Franz DT, a hearse moving things from one side to the other. Now, the idea is that we want you to take the girl. Sorry. The idea is that we want to take the integral of those sites, just solve our friendship, create That's how separable equations work. But to take the integral off this left hand side, we have to right this fracture as a sub off two fractions is in the method of partial fractions. Let's quickly recall how that's done. They have this fraction here. Sorry. You have this fraction here. One over. Be right. It's your fifty. Sometimes the line a seven. You want to write that at some constant over feeling, Yusuf. If it worse so other constant over feeling a certain fish. No, hide that Well, let's simplify this equation by multiplying through Bye. This the nominated If we multiply through by this, we get one equals. Now. Here. You already have a few minutes. You fifty. So you're only left even I have a seven fifty and he already have a people in seven fifty. So you only left for Beaver and it's your fifty. So Carly, figure a V where we can just plug in specific Clyde is for P and we get equations which we could himself. For instance, if we plug in because seven. Fifty, then seven. Fifty seven, fifty eight zero. So the coefficient of a here dies off then seven fifty minus two. Fifty is five hundred. So your occasional reed one equals eight and zero for speed sci fi ve And this means that be because one over five and similarly you can figure out by plucking and, uh, two fifty. Instead, the exact same thing will happen, except that this time we will have two. Fifty minus seven fifty. That's negative. Five. So when you work out a way, you would get Mega Kuzmin five hundred. So that's how the method of partial fractions work. X don't. So now let's go back to our integral. So we want you integrate at this integral ahead Here on the left hand side, we can write this as, uh, enter Grew off. So right, We have a over B minus two fifty. Where is negative? Another five hundred. Pull us over fifty first. And then he was one of five hundred Brian Speer, all of that. Now that these Constance one of the five hundred here and here, I can just factor out. So I'm gonna have, well, five hundred here first and then the one over even issue fifty. The integral of there is just ah, log. So we have just log off the absolute value off people's shit. Don't forget the night that sign we had here, right? Because of this night. The sign on the one of the five hundred and then But the indigo off one of the people. And seven fifty. That's his drug B minus seven. Fifty. All right, plus the constant. But this constant will show up spec plus a concert here. So the other integral was easy. This is just the integral off constant. So the integral off so and the girl. But this blue intergroup ofthe night of Syria's is eight e. R. This is zero eight baked B. It's a mother constant. Olek. Yes. So it can equate this. We have obtained here that one. Can I quit this with this right? That's what we get by taking into go off both sides. So let's do that. So we get no that. Let's see. On one side we had a longer he went seven fifty Dryness Log off, even Sue fifty. Right? If you look back to this side, let's ignore the one of her five hundred for now. And because we're going to move it over to the other side by multiplying through by five hundred. So he had Lord be one seven fifty minus log minus two fifty equals. So since we're multiplying through by Constant, uh, five hundred, we had a class. See that? Because it's five hundred c that will eat crow. Ah, on the other side, we had five hundred times and this year's ears is age t first. Thanks. Now it can just finish multiplying this. So here on this side we would have our people in a seven thirteen planets Park Su fifty and the rats move this five hundred c over to the other side. So we have ah, five hundred times this work cell tonight O forty Ah, first five hundred B minus five hundred six. Now this favor thie minus five hundred. Sea is just some other constant. All right? He's just a non viteri constant. And see, just a lavatory costed. So if you could just give this another name, I'd say e So we can write this. We can write this as, uh, that's it. Who are the seven fifteen minus far Sufism? Because I go for a pee breast some constantly. I doesn't really matter that we will define it by five hundred orders. I don't know if you recall a log rules. A difference off launch is the log off the cautioned seiken. Right, this is log off. So, Christine being right is to first because But the prosecution take this by using a long cruise. No, What can I do? I can just to get rid of the law, I can just expert in shape both sides. And so what happens if expert in shape both sides of peace that this equation well on the left hand side would just get rid of the Lord? Be Afghanistan, fifty over. You know, Sue first the right hand side we get now. Let recorded an exponential off. A soul is the product of thie exponential. So first you have it's exploding. Show Grimes, you have support. Transfer was near, raised your capital. Now this theory should capital E. We can just meet again. It's a constant since Jesus and lavatory cost. So let's rename this constant cape and let's right. Finally, the B minus seven fifty over being with Sue fifty because a constant K fine be regular. This is what the formula for our solution looks like this. No. Okay, so now the question asked us to use initial populations of two hundred and three hundred to graph what the solutions. So that's sort of two hundred. So let me rewrite this here. We're looking keeping in mind. So I had the one seven fifty over B minus your fifteen because a constant me to the negative four. All right, so now let's suppose that we start with a zero okay, off two hundred. That's reason about these absolute values. Now, if I start over published of two acted and two hundred minus seven. Fifty is leg five. Fifty and two hundred minus two. Fifty is negative. Fifty. So both the top and the bottom are negative. And when I defined night by negative, I get a positive. So since this finger, I'm getting responsive. Anyway, I don't need the absolute violence sighing and I can just right without absolute fact is people in its own fifty performer give minus two fifty vehicles. Kay needed a night off for tea. Now let's figure out what the constant is by just plugging in two hundred. Right? So we plug in time equals zero time. It goes zero and a population equal to two hundred. So if Buzz we should plug that in tow, our equation we get here two hundred minus seven. Fifty over two hundred, my sou. Fifty four people. Okay, then here. I'LL have night of O four times zero zero and zero zero's Just what? So our case is thiss right? And if you work this math out, you will see that she get kay. It could happen. So they have k equals eleven. Then what can we do? So here we can write this as eleven isn't the whole forty. You know, Katie was eleven now. We haven't equation here with our peace on one side. And this in the other cell we can multiply through by B minus two fifty and then solve for Pete. This is just gonna work out to be a simple linear equation. Which MP? So you can solve this for B on. Do not be too much of your time. If the answer is B equal seven fifty miners twenty seven, fifty feet of the dragon for De decided by one of minus eleven. He didn't, for this is what you get as your answer. That's Blissett, Forman. And so if you graph this using our couch earlier and tried to that appear if you graph this is in your account, clear. A little practice at Syria. You started two hundred and then it would decrease rapidly to zero. So this point here is that so. It starts at two hundred and the crazies weapon reaches. Here you have two hundred and you can check numerically that it reaches zero at about thirty two. Thanks. The bridge is your but this is more or less what the graph of this will look like. You just use your calculator now, which is basically what we expected from Hard drone, right? A population of less than two fifty like two hundred who just decrease rapidly to zero and will be zeroing in on family life. So now that what let's do the math for what happens off population above two fifty, like three hundred? Well, we again start over formal of P five seven fifty over the two. Fifty. Because, Kay, I needed that Pegasus for the no. Uh, now we have to reason about the absolute Find his things carefully because no b zero, we're going to be easier to increase it. So if you put three hundred here, three hundred minus seven fifties night before happened, but three hundred minus your fifties fifty. So I get a night of over pasta, which is negative. So to really get the absolute value I need to flip the sign, right. So really gets something like, uh, negative. Meanwhile, seven. Fifty over him when sue fifty. Because, Kay, you know, forty. But to figure out what K is liken displayed in T equals zero and people's three hundred as before, this e to the whatever will become one. And I just have a formula for Kate. So if you do this, you will find this time Jake? Because Randall Yes, you'll find Keiko's night. Right? So given that cake was nine, I can write this. Oh, that's because mine needed the forty. All right? And then I have a linear equation here on one side. I have my piece on one side. I have my eating bananas over forty. I can multiply through by p minus you fifty and ourself this equation for B. And if you do this, you will get a phone for Be very similar to what we had before. But this time I get talent. Fifty plus three. Fifty. You did that. D all of that decided by one one Verse nine. Eat it. Oh, is your funnel for P to work out this simple Ah linear equation as before? So I should graph this using powerful here It's down here. What we have it would look something like this. Your population will start at three hundred at st three hundreds around here and grew It will change from cavity at some point and we'LL have seven fifty as a horizontal acid. So it will start at three hundred and it will have seven. Fifty as Horace until ass in here the uh, you'LL never actually reach seven fifty. If you work out the living off this expression of Steve goes to infinity, you will see that you indeed get seven. Fifty. Since this far here goes zero on this part. Here goes zero. So you only left of seven. Fifty over one to sell fish. So that's what the explicit equation for the solution looks like. And it behaves exactly as we had expected from our drawing off the directions for you. So that's it for this question. Thank you. Have a good thing.

All right. So this problem asks us to suppose that a population grows according to a logistic model with carrying capacity six thousand and K equals zero point zero zero one five prayer. So part ay, they want you to write a logistic differential equation for this data. Well, this over here is our general form of this differential equation. Right. Well, we have that Kay's points here zero one five and M, which is the carrying capacity of six thousand. So this is how we can right this model here using the logistic differential equation. So there we go. Heartbeat now wants us to draw a direction field so you can do this by hand, or you can do it using the computer after system, I used the computer out of the system s o. Basically, all of these lines are telling you that if you had a point at any one of these points, that the way the vector is pointing is the direction that the curve would have to go. So you can see here that at six thousand, because we have basically a flatline here at six thousand. We see that we haven't equilibrium point here meaning that once the population hit six thousand, it's going to stay at six thousand. So if we start anywhere below six thousand, are functional increase until it hits up to six thousand. So taper off as it gets close to six thousand. On def, we start anywhere above six thousand, our population will decrease. I want skin taper off and converge to six thousand. We also see here that our population, which is is the vertical value here, does not depend on t meaning that if you pick any height, any population and he look all the way across well, all of those directions are pointing in the same direction. So if you move horizontally, that won't affect the direction here on the direction. All right, Part C asked us to draw curves on this direction field for initial populations of one thousand, two thousand, four thousand and eight thousand. So the one in red is with the initial population one thousand, and we can see here that the solution is Kong cave up for a little bit, which means that our population is increasing and increasing Richt and then around about ah, one thousand for tea about right here, we see that we have inflection point and then our solution looks Kong cave down after that. So our population is still increasing, but at a decreasing right, which makes sense because we need the population to taper off at six thousand at two thousand. It's concave up for a very short period of time and switches to con cape down once again to taper off as we get near six thousand. If we start with a population of four thousand, you can see that we're calm. Keep down everywhere, which means our population will increase at a decreasing right. If you look at eight thousand over here, you see that we're con cave up everywhere. So our population is decreasing because we need to taper off at six thousand but is decreasing slowly. It's it gets pretty steady and papers off at six thousand here. So these inflection points that we saw mainly with the initial population of one thousand, is when our population changes from increasing at an increasing rate to increasing at a decreasing right. All right now, part Dean asked us to use Oilers method so that you could program a calculator or a computer to use Oilers method with step size. Age equals one to estimate the population after fifty years if the initial population is one thousand. So, um, this formula over here is from section nine point two, I believe, and this is Oilers method in general. So if we have a differential equation, that's this Y part equals affects where they have X. We used tea in this problem and where they have why we used P. And so this is our initial condition right here. And this is how we're going to define our exes a CZ we keep going up. So for each value of X, we add the step size to it. In our case, that will be one on DH that gives us our new axe value. And so we get our new Why value Why? End is what we take the previous why value and add to it the step size So one in our case, times f of the previous point. So I have here some suit, a code for us. So where you want do this on the computer? You calculator. This is the general outline. So your first step? Well, we need to define our function, and that's going to be our logistic model that we did in part A. So you already know how to get this equation. Here are second step now. We need to input our initial start time and our initial population. So we wanted to start with the population of one thousand. So we have p zero equals one thousand here, and we're starting that at time. Zero. Next, we have to input our step size. So it's a Chico's one, and this an equals fifty here. It tells us that we're going to do this or into fifty steps, which makes sense because we need to get from your zero to your fifty to figure out with the population is after fifty years. All right, so apart for here is really important. And it's where we actually execute oil. Is that so? We're going to do a four loop and we're gonna do this fifty times, So from one to fifty. So the first part of this loop is Well, we need to figure out what f of teaser and P zero is, and I'm just going to call that a Well, then we can see our next population will approximate our next population with Oilers method in part be here. And so we're gonna take our initial population and add to it or step size times. Well, what we found in party here, then we're going to increase our time. So we're gonna say t one is the time we started off with, plus our step size, which is one in our case D here recently want to print the new time and the new population is so we can keep track of each step of the process here. And then Parts E and F here are redefining t zero and P zeros that we knew this loop again and again until we do it fifty times. And the other day, we need to end this program. So if you run this, you're going to see that after fifty steps, which means fifty years, the population well, this is going to spit out one thousand sixty four point zero three nine on and on and on. But we're talking about population here, so we want to talk about whole numbers. It doesn't make sense to talk about point zero three nine, but person so well, actually saying this gives us roughly one thousand sixty four as our answer here for using Oilers method. All right, so now Part E wants us to actually find a solution here if we have an initial population of one thousand. So that is what we're doing here. Thiss is our initial population. We're sorry. This is our solution to this differential equation and that comes from this section in the book. And so and once again, our carrying capacity is going to be six thousand. And this depends on this. A here. Well, a is going to be a carrying capacity minus our initial population, all divided by our initial population. So moving over here, we can compute that within the framework of this problem. So carrying capacity, this here is our M. And then here's AARP easier. That's one thousand and a ends of just being funny. So this here is our solution. It's a population function. So we get p of T equals six thousand over one plus five times E to the negative zero point zero zero one five t. And this part of the question also asked us to figure out what the population is. B after fifty years using this function so you can use your calculator here and you'LL see that once again we get roughly one thousand sixty four, which means that orders not it was pretty accurate. All right, so my late part f wants us to actually graph this function that we just found. And so this here on the left hand side, there, that is our graph this one over here of this function pft that we found in party. And so it also tapers off at six thousand. That's what this line is here. And you can see it has an inflection point roughly here where we go from Com cave up. So the population is increasing and increasing rate to Kaan cave down. So our function is still increasing, but at a decreasing right, because their populations why do taper off six thousand, which was our carrying capacity? Ah, and the graph I have here on the right is the graph that we drew using our direction field. So the red line on the graph on the right was also with the initial condition one thousand. And if we compare the red curve on the right to the red cover on the left, they look like they match up. So it all worked out

However, what you are going to school Problem number Mine absolution Be off the equal zero on a B T quote d cavity so we can get slope at be great Roseanne T capital. So deep B boy de de represent zero on that. Be listens nt on grid was and judo. So DP boy the D lists and zero. So thanks to you can get the eyes of planes, So Oh, what? Deploy Be negative. The equal. Okay, so he squirt finest deep be minus. Kate over. Out. Equal zero Hence be 1/2. Deploy d was a minus square root. Are these words bus for K? Okay, Over. Okay, so as we see, that's global. That solution sets, Boyd said, spoils the condition k separated or equal Negative are his words over four. Then we think it's the ability. So he s court be Bye bye, Q squared we are must employ to be in that of the d B by DT. That's equal our squirt r squared. Oh, like to be in like a d being negative. D off the fly be. And the solution curious Bar conch up. Uh, four. That's for B. Waiter San Torto and couldn't give down for be lists that they cover over tea on the greater than zero and they touch it. Figure will illustrate. That's for where to seize a T access and fealty access answered the cavity for number C four four. Be note graters and dear want lists and D that population population dies, dies out with time and for be note beyond feet. So he notes, Great present T capital. There is a population roast, so the term off the shoulder level is accepted since D gives many months. Oh, you know about which there is a population gross sex for watching.

So for this problem we have the Census Bureau and we know that um the estimated growth rate of K of the world population will decrease by roughly uh 0.2 per year for the next few decades. So in 2000 and four, K was 40.132 So we want to express K as a function of time. So we see that K um equals originally it was 0.132 And it's a decreasing at a rate of 0.2 T. So you see, it's decreasing over time. Um since 2000 and four were t equals zero. And we could find a differential equation that models the population for this problem. Um And we were just multiplied this whole thing right here because this is the rate we would just multiply this times. Why? That was the R D Y G. All right. There is going to be our final answer. And then we could solve this differential equation and graph it using the methods that we've discussed earlier.


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Let flr) = 9(5)*-s . Evaluate f(6) without using a calculator.Provide your answer below:f(6)
Let flr) = 9(5)*-s . Evaluate f(6) without using a calculator. Provide your answer below: f(6)...
5 answers
Vz 4 Y- 2 Jx, X-O , 128T 4 4 D 2561 2 C2s61 1 D. 1284 E,%
Vz 4 Y- 2 Jx, X-O , 128T 4 4 D 2561 2 C2s61 1 D. 1284 E,%...
5 answers
For each ofthe following pairs of alkenes; indicate whether the members of each pair are positional constitutiona isomers Of skeletal constitutional isomers_(a) 3-Methyl-Z-heptene and 3-methyl-3-heptenepositionalskeletal(b} 2-Methyl-l-octene and ~methyl-I-octenepositionalskeletal3-Methyl-Z-heptene and 2-methyl-Z-heptenepositional skeletal3-Methyl-2-heptene and methyl-3-heptenepositional skeletal
For each ofthe following pairs of alkenes; indicate whether the members of each pair are positional constitutiona isomers Of skeletal constitutional isomers_ (a) 3-Methyl-Z-heptene and 3-methyl-3-heptene positional skeletal (b} 2-Methyl-l-octene and ~methyl-I-octene positional skeletal 3-Methyl-Z-he...
5 answers
Find the area of the shaded region.(Diagram Cant Copy)
Find the area of the shaded region. (Diagram Cant Copy)...
5 answers
∫ sec 6 x tan x dx
∫ sec 6 x tan x dx...
5 answers
The reaction between hydrogen gas and nitrogen gas producesammonia gas according to the reaction: N 2 ( g ) + 3 H 2 ( g ) → 2N H 3 ( g )What volume of hydrogen gas is needed to react with 17.7 L ofnitrogen gas if both gases are at STP?If 22.3 L of hydrogen gas reacts with excess nitrogen gas atSTP, what volume of ammonia gas will be formed?
The reaction between hydrogen gas and nitrogen gas produces ammonia gas according to the reaction: N 2 ( g ) + 3 H 2 ( g ) → 2 N H 3 ( g ) What volume of hydrogen gas is needed to react with 17.7 L of nitrogen gas if both gases are at STP? If 22.3 L of hydrogen gas reacts with excess nitrogen...
5 answers
What is DNA's structure described as?a. a step ladderb. a double helixc. a tertiary protein-like structured. barber pole
What is DNA's structure described as? a. a step ladder b. a double helix c. a tertiary protein-like structure d. barber pole...
5 answers
21.(8 pts) Propose an efficient synthesis for the following compound, starting with CH,CHC-CH and BCHCHs.CH;CHzC-CHCHCH;
21.(8 pts) Propose an efficient synthesis for the following compound, starting with CH,CHC-CH and BCHCHs. CH;CHzC-CHCHCH;...

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