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(10 points) The projected rate of increase in enrollment in a nCW college is estimated by dE (at + 1)-(+1), t20, dt where E(t) (thousand of students) is the pro...

Question

(10 points) The projected rate of increase in enrollment in a nCW college is estimated by dE (at + 1)-(+1), t20, dt where E(t) (thousand of students) is the projected enrollment in years: If the enrollment is b (thousand of students) now (t = 0). find E(3). the projected enrollment 3 years from now:

(10 points) The projected rate of increase in enrollment in a nCW college is estimated by dE (at + 1)-(+1), t20, dt where E(t) (thousand of students) is the projected enrollment in years: If the enrollment is b (thousand of students) now (t = 0). find E(3). the projected enrollment 3 years from now:



Answers

The projected rate of increase in enrollment at a new college is estimated by
$$ \frac{d E}{d t}=5,000(t+1)^{-3 / 2} \quad t \geq 0 $$ where $E(t)$ is the projected enrollment in $t$ years. If enrollment is 2,000 now $(t=0),$ find the projected enrollment 15 years from now.

Welcome to this person. So there's a question represent their reach of attendance. The rate of enrollment students with respect to time in years. We are looking for the number of moment. So in this sense we would integrate this so that we can get the number of enrollment is not having the rich. So this is a call to the integral both sides. Mhm. Yeah respected. T. I will make a substitution one Lazar point to So that you have the you that is a call to 0.2 18. Then I would have the T cost too. Do you all the buzzer point to so your substitute this and you substitute that. Well we have the number of war mates. But a sequel now in you. Yeah, So you have 2001 plus Their .2 T. Now becomes a you have 30, The battery in two. So the T becomes 0.2. Oh okay. So instead of to put in, do you All over 0.2. So here we have and a few weekends Let me see, called to come bring 10,000 outside. That is 2000 and writer writer by two. You have this. Yeah. So if we integrate it 10,000 that we have you The behalf all over one. All over negative last see So this becomes 20,000 negative 20,000 you to the bar. Classy So you can write that negative 20,000 all of us crimes of you then blessed. See now we can like a written in terms of T. So negative 20,000 All of us carried of one class. They were .2. T. The glass see no. Currently what is called to there we have 1000 in rooming. So I was in their roommates. That is correct. Of one plus 0.21 It is going to zero. Yeah, When she's called to zero we have 10 days ago. 2000. All right. So let's see initial body. So this becomes 1000. That is equal to negative 20,000 blast. See Now this is one skirt of once or makes it point At this point we can add 20,000 to both sides of the of 20,000 blast 1000. She called to see soc support to 21,000. All right. Then you can write and of G. S. Negative 20,000 all over Skirt of one other point to T The last 21,000. All right. Now you're looking for The woman went is called to five. So at T because 25 Yes and of five and that is she called -20000. Well that was kind of one last year. Going to increase of tr put five there mm and secure to So that is negative 14 14 2.13 56 Yeah. Mhm. So if we subtract Would have 6857 0.86 You're looking at a number of people. So a pause mainly. Mhm. We have 6858 students. Alright. Some short time. This is the end of the lesson.

So here in this question we're gonna evaluate uh by using indignation. So you're supposed to do this question. We were supposed to find an expression And giving the enrollment of 80 years from now and also the environment five years from now. So this is the expression given here. So I can write this as this is a variable and that is dependent on T or time where n is the number of students enrolled. So again write it like this One plus point to the race to -3 x two. I just changed and dusty to be invited to nothing else. So here what I'm going to tell you is that we've been doing in division by substitution. So we're going to substitute this particular term as you. So then what would happen is one plus .2 can be written as you. So by differentiating this term will be getting 0.2 day T. This would be zero because that's a constant is equal to be. So instead of tea, I'd be substituting this whole term with you. So D and by D will be coming. So before that I'll just show you that I'm gonna rearrange this particular expression by taking being on one side and data on the other side. So then this would become B. N. Is equal to 2001 plus point to be traced to minus three by two DT. So this would be the expression now we'll be replacing this particular tv have DP we have to do. So what would happen is DT would be nothing but do you by 0.2. So this will be the change. So strong beauty. I could use do you by 0.2. So then the changed things would happen here. I just write it down so then I can write B. N. Is equal to 2000 instead of one plus point totally. I can write it as you You raised 2 -3 x two. It's of data and I'd be you by .2. Do you by 0.2. So you can cancel this thing and this thing that would give us 10,000. So then I can write D. N. Is equal to 10,000. You raised 2 -3 x two. b. uh b. So what happens here is I am supposed to find the value of anywhere in is the number of enrollment of students with respect to pee. But I have changed the variable T. To you. So now what happens is so now I'm gonna integrate on both sides because I've got a small change of N. With respect to a small change in you. Now I need the whole value of N. So I'm going to integrate this both sides. So then I'll be getting And be an integral begin would be N. is equal to 10,000 is a constant. I can keep it outside. You raise two minus three by two integral of U. S. Ministry by two years. You raise too. And it's similar to integral x rays to enrich his x rays to end plus one by and plus one. So that will give us you raise two minus one by two by minus one by two. And uh this will be done. So we'll be getting So you raised 2 -1 by there's nothing but one by routine. So I plus the ribbon integration Constancy. So then N is equal to I can keep it here -20000 by route to you. Plus integration Constancy. So this is the expression which I've got with respect to you. So again re substitute the value of you in terms of P which is one plus 10.2 P. So then we'll be getting in which is the number of students central is equal to -20000 by root tea. Is can try to use equal to uh we have here one plus point ot so I can write it here one plus 0.2 T plus see So this is the general expression which I've got. So for any time t this would be the value of him. But we don't know this constancy. But we're already given an initial condition that at physical zero. Initially the number of students central is 1000. So we can take that particular condition here to get the constant C. So at P is equal to zero 10 is equal 2000. So we're getting 1000 is equal to minus 20,000 by physical 20. We're getting at route one plus C. Which implies C. Is nothing but 21,000. So I can further right this particular expression here to give I'll write it in green. N is equal to minus 20,000 by quota. One plus point to T Plus 21,000. So this is the general expression this is one of our question. This is the general expression will be getting in terms of an anti for any time. T. This would be the number of students enrolled. So from here we'll be answering this is one of our answer. From here we can answer for let's say it is equal to five. Business need to substitute. T. So what would be end and would be nothing but minus 20,000. Bye. I'll write in some other color and will be And is equal to -20000. Bye wrote a one plus .2 Window five Plus 21,000. That would give us -20000 by route to Plus 21,000. The city calculator you'll be getting 685. It to be to answer. So this is the second answer which we have to find. So this one is a general expression at any time. T. The number of students and problem with this. And this would be there number of students enrolled at physical to five

So we have the equation. A a T equals 29,000 comes 1.3 to the T. And so we want to know when this account is gonna get to 35,000. So, really, 35,000 beer A of t. Now, the first thing to do is divide by that 29,000. That's gonna give us 1.207 equal to 1.3 Raise to the T. Now we're gonna use them logs, and we're gonna take the glogg, both sides. So we're gonna do the log of 1.207 equals now the tea can come down. Um, power rule says the exponents can become a coefficient times the love of one point of three, then divide by that wall. And so we can determine that this will happen in approximately 6.4 years. In Part B, we want to know the double income so troubling 29,000 would make 11 another when it's going to get to 58,000. So dividing out the 29,000 we get to equal to 1.3 to the T longer. Both sides bringing down the tea with the power rule dividing by the long of one point of three. And so when we do the log of two divided by the log of 1.3, we got approximately 23.4 years to double this money.


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