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(10 points) In many population growth problems, there is an upper limit beyond which the population cannot grow. Many scientists agree that the earth will not suppo...

Question

(10 points) In many population growth problems, there is an upper limit beyond which the population cannot grow. Many scientists agree that the earth will not support population of more than 16 billion: There were 2 billion people on earth in 1925 and billion in 1975. If y is the population years after 1925 appropriate model is the differential equationky(16 - y).Note that the growth rate approaches zero as the population approaches its maximum size_ When the population is zero then we have the

(10 points) In many population growth problems, there is an upper limit beyond which the population cannot grow. Many scientists agree that the earth will not support population of more than 16 billion: There were 2 billion people on earth in 1925 and billion in 1975. If y is the population years after 1925 appropriate model is the differential equation ky(16 - y). Note that the growth rate approaches zero as the population approaches its maximum size_ When the population is zero then we have the ordinary exponential growth described by y 16ky. As the population grows it transits from exponential growth to stability: (a) Solve this differential equation: y (b) The population in 2015 will be y (c) The population will be 9 billion some time in the year billion_ Note that the data in this problem are out of date, so the numerical answers you'Il obtain will not be consistent with current population figures_



Answers

The Census Bureau estimates that the growth rate $k$ of the world population will decrease by roughly 0.0002 per year for the next few decades. In $2004, k$ was 0.0132 (a) Express $k$ as a function of time $t,$ where $t$ is measured in years since 2004. (b) Find a differential equation that models the population $y$ for this problem. (c) Solve the differential equation with the additional condition that the population in $2004(t=0)$ was 6.4 billion. (d) Graph the population $y$ for the next 300 years. (e) With this model, when will the population reach a maximum? When will the population drop below the 2004 level?

Okay, so we're given in year 16. 50 we had a population of 500 million, and in year, um, six or 2010 we have a population of 6756 million, and we're asked for part A to find an equation for the world population. If you grow exponentially, that is, we have a is equal to PRT. Actually, it's instead of a we want pft. Okay, so assuming that 16 50 is, um, time or time starts at zero years, then you know that the time for 2000 and 10 that's equal to 2010 minus 16. 50. So that gives me, um, 360 40. Okay, so let's plug in. Um, that's years old. We have 500 million. That's equal to p. And then we have I need to the power of our time. Zero that gives us that RPI value is equal to 500. So we have that plt is equal to 500. You need to the party. Now. Let's plug in year 2010 that has a population of 67 funds is equal to 500. You to go are times 3 60 Now we're asked us all for our to solve for our population inclusion. So 676 divided by 500 gives us 13.5 or two is equal to be sued. A power of 3 60 times are and I would say core natural log both sides to bring down that policy. You get 3 60 times are and an arm is equal to a one of the falling over 3 60 and thats approximately That's approximately zero point cells. They're all 7 to 3. Okay, so our equation for population, if it grows exponentially, is you have to use equal to 500 bto 0.0 07 to 3 to the power of T. Okay, I know we're asked for part B to find, um, our population in the world at year one. So that's p evaluated at one is equal to 500 e tra 0.723 times one. So what does that give us? But see points, although 7 to 3 times one yet 503.63 1,000,000. Okay, so we see that at your one, um are difference from 500 million is just um, 3.63 So we see that at the beginning are starting years we grow slowly and then we increase exponentially.

We are convinced that be off the equal 73 point toe upon 6.1 plus 5.90 days toe buying a 0.0 duty when the equals zero years 2000 on that every year I didn't do it. So the year 2200 is going to be d equals 200 the year 2300 is gonna be Do you implicitly we go toe the I 84 calculator and blood are function read. I cannot function so that's them Now we tap on the glass on Now we have to find the value for X equals 203 100 We go toe the Cal commando on the select value and their eggs equals 200. That gets us value as 11.79 billion on for X equals 300. That means proceed do we get the value of an X equals 300 as 11.97 billion. So we know down that in here 2200 the population with 11.79 billion on the year 2300 the population will be 11.97 billion for Part B. We have already drawn the craft for the years 2000 to 2500 so that is done. A now for part C. We can clearly see that the population model approaches this point using the graph so we go and knees do mode on, Gentry said. When times increasing the population grudges and it converges with value off good so the population will end up conversion at 12 billion.

So in this case, we're looking at population growth and were given a function here that talks about the popular the world population. Okay, so P of t is equal to this function here when t zero we're looking at the year 2000. So t represents the number of years after the year 2000 and the population pft will be represented in billions. So my first question says, Okay in 22,200 what would we predict? That the world population will be Well, in this case, that means I'm going to evaluate p of 200. When I evaluate p of 200 I'm going to get 11 0.79 billion people. Then it says to evaluate. Well, what about 4000? 300 year 2300 Which means I'm gonna put 300 in cause 300 is 300 years after 2000, and that gets me 11.97 Well, if we sketch a little bit of a graph here will notice that the graph is leveling itself out. The big question is, where is it leveling itself out? Well, based on these values, right here, we can kind of take a guess and you will see that it is leveling out close to 12 million people. So what this means is, as time moves forward, we should expect that the world population will level out at 12 billion people.

Since pulled hard in the world. Population in 1,000,000 Close lifted the excellent Sir Vincent. It is given a why Equal toe six genital 79 You devour judo point genital 1 to 6 x where X is made from it went eggs is the number from here since boot. Ardent. Okay, No. The first. The world population waas 655 million in Portland six support food Italian six x will be six So why 6.79? You do our judo 0.16 into six which leads store 655 6.4 So it is close to it is close to the given 6555 1,000,000. But what part? A be use the model s trade the populace input within 10 So for food Politan X equal to 10 6079 Beautiful judo point Joe one through six and execute took 10. So in total then it is Quito 68 95 1,000,000 population. Let's see in both ardent Okay, You need smarter to predict in 2015. So that spoon fire 25. So why equals six Geo 79 and it'll Algeria 1016 in 25 which is equitable. It do 839 1,000,000. It's see. It is increasing exponentially, but population is not increasing exponentially, so it may not proof or a create for 25 the population dozen arc in Crete exponents silly therefore hit this mortal may not equity part both hardin friendly fire and they did answer.


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