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60e0.237t years after 1950 is given by P(t) Find the The population = of a city; in thousands, percent: Round your final answer (in percent form) continuous growth ...

Question

60e0.237t years after 1950 is given by P(t) Find the The population = of a city; in thousands, percent: Round your final answer (in percent form) continuous growth rate_ Write your answer as decimal place_ Do not put the percent symbol in your answer: to one

60e0.237t years after 1950 is given by P(t) Find the The population = of a city; in thousands, percent: Round your final answer (in percent form) continuous growth rate_ Write your answer as decimal place_ Do not put the percent symbol in your answer: to one



Answers

The population of the United States was 3.9 million in 1790 and 178 million in 1960. If the rate of growth is assumed proportional to the number present, what estimate would you give for the population in $2000 ?$ (Compare your answer with the actual 2000 population, which was 275 million.)

All right, So for this question, we can write p of seven. Pier seven is equal to Pierre. Zero times s 07 plus the integral from 7 7 0 07 0 to 7 off, one over eight, minus teatime. 1600 dt. So, uh, we simplify That's down a little bit. This ends up being 36 1000 divided by eight plus negative 60 100 times ln off the absolute value off eight minus t on. We're going to evaluate this from 1000 8 60 100 8 0 to 7. So this is 36,000 divided by eight plus 3327.10 And the enter to this is approximately equal to 7827.

All right, we know that this right here is the population growth formula were asked to find the rate of population growth, which is k constant for Arizona. Between 2000 when the population was 5,130,632 2 2006 on the population grew to 6,123,000 106. So what we need to do is plug these two values into our equation and calculate t and then solve. So why is the population we ended up with? So why is our population in 2006 and that equals Why not? Which is our starting population? Times E to the k times t now t is time. So between 2000 and 2006 that was six years. And then kay is our unknown value. So we're looking to solve for an unknown exponents, which means eventually we're we're gonna take a log rhythm. The first thing we need to do is isolate this exponential equation, and we do that by dividing by our coefficient. So we're gonna divide by 5,130,632. It this becomes one we can calculate. This is gonna be a decimal in our calculator. So six million 123,000. 106 divided by 5,130,000 632. Whoops. Try that again. 6123 106 Divided by five 130 You 6 32 Yeah, we get a value of 1.19 and that equals E to the six k now, because we haven't unknown exponents. Like I said, we need to take a log rhythm. Now we're working with E. So the base e is the natural logs we need to take. The natural log of both sides of the natural log of 1.19 equals the natural log of each of the six K. Remember taking the National League of both sides because with the equations, what you do, the ones that you have to do to the other. So on the left, we still have the natural log of 1.19 and on the right, we're gonna use properties of exponents to bring this six k down in front to multiply by the natural Aga be. So any time we haven't exploded within our longer than we can bring that down and turn it into reread it as a multiplication problem. Now, look here to simplify the natural log of e is one, so that quantity is gonna go away. And now we need to get Kay totally isolated. Right now. It's being multiplied by six. So we're gonna divide by six on both sides to solve for Kay. We're solving to the nearest tent. So we're gonna use our calculator to do this so K equals the natural audible in 0.19 divided by six. So let's go over to the natural log of 1.19 and we're gonna divide that by six, and we get a value of K of 0.28 or to nine with me. Round K equals 0.0 three. And that will be our constant population growth in Arizona between 6 4000 and 2006.

Welcome to this lesson. In this lesson. We are looking at the rate of change of our population after they would find the population for the current year. We use that to find a population in five years stuff. Yeah. Yeah. Oh yeah. Who would take the into girl on both sides in order to find the population. So the population will be called to the integral. Whoa. Uh huh. This one will be very great if we split it so that we have in this way and okay So straight the way this becomes 400 T. We are left with this one. And this is what we do. We would equidad you to 20 for glass T. Squared. So that you would become to T. 50. Then we have the DTs do you on two. So place this into the equation. Now we have the P. T. The population at the time T. Cost 400 T. Plus the integral. We have 800 T. All over 24 Plus T. Squared. So that represents you. And now we have. Did he? That is a quote. Do you on to T at this point this tea would cancel. That's So that we have 400. Because the two also council South's new d. You Then this is 400 t. The last the whole thing becomes 400 lan. You classy This becomes 400 t. The last 400 Land. Yeah. Whoa That is 24 plus cities spread and plus you. That's when we can use the initial value given to us to solve for the value of C. The population was 60,000 when the time was zero. So p of 0 60,000. So we have 60,000 for the sequel to 400 418. Land 24 plus zero is quite so like 24 last summer. So 60,000 would be caught too. Uh huh. Okay. Okay. Oh That is 1 2 71 0.22 glasses. So C becomes 60,000 minus 1 to 71.22 So that the sea becomes 58-72 8.78. All right, then we can have the whole thing which would be yeah, 24 blast t squared. Then the last 6 8.7 uh 687 to 8 point 78 Tonight. Let's look for p. five p. 05. Oh. Mhm. Okay. Oh 24 glass. Can you four blasts any five? Oh. Uh huh point so P five. So that's the population and that is paul 32 nine. All right. Since the time, this is the end of the lesson.

So this question we are supposed to find the population from five years from now. They're given the population of a certain city which is product projected at raped R. Of T. Is equal to 401 plus two T by 20 for plastics. Well So we are given an initial condition at physical zero. The initial population 60,000. So we can write this thing as D. R. By D. T. There are is the population with respect to T. and years is equal to 400. one place to be by 24 plus T squared. We can rearrange this thing to give. Mhm. The R. Is equal to 400. I'm going to multiply this here. So 400 DT plus 800 p By 24 plus the square DT. Now if you try to integrate on both sides try to integrate on both sides. Be getting R. Is equal to 400 B. Plus. I'll just take uh 400 outside 400 integral. It would be why 24 plus the square did. So now this is difficult and degrade at this particular moment. So now we are substituting at this moment. It is assumed that 24 plus these four is equal to you. So then two DT is equal to do this particular term would become the you. So then I can write R. Is equal to he's black. So our would be equal to 430 plus 402 integral DT by you. So we already know integral. One by X. Dx is nothing but natural logarithms explosive division. Constancy. So that would give us our is equal to 400 p plus 400. That sort of longer than all for you. Plus integration agency. So now I can re substitute the value of you to be 24 plus the square. So then our would be nothing but 400 T plus 400. Natural log off 24 Plus T sq plus immigration agency. Now we are already given an initial conditions that are is equal to zero. The population is 60,000. So you're going to use this particular condition that the single zero or a single 60,000 in here to find the value of C. So that would give us 60,000 is equal to 402 0 plus 400 natural algorithm off 24. On calculating this will be getting the value to be C plus C. So by taking the old ones on other side and keeping sea on one side reading the value to be +587 to 8 0.7. This would be the value of C. So now substituting this particular value of C. In the situation which is here we'll be getting R. Is equal to 400 P Plus 400 nuts or locate them off 24 plus the square plus 587- 8.7. So this would be the general expression at any time. T. The population would be this. So now we were supposed to find At T is equal to five. So here we just need to substitute so we're getting our is equal to 400 into five plus 400 natural order them off. 24 plus five squared is 25 Plus 587- 8.7. So that would give us five and before too 2000 plus 400 knights of local them off 49 plus 5 +82587 to 8.7 on Further simplification will be getting The value of R is equal 6-8 5 .5. This would be the final answer would be the population at T. is equal to five.


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