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The rat population in major metrol politan city is given by the formula n(t) 25e"04 where t is measured in years since 992 and n(t) is measured in millionsWhat...

Question

The rat population in major metrol politan city is given by the formula n(t) 25e"04 where t is measured in years since 992 and n(t) is measured in millionsWhat was the rat population in 1992PreviewratsWhat does the model predict the rat population was in the year 2001Preview rats

The rat population in major metrol politan city is given by the formula n(t) 25e"04 where t is measured in years since 992 and n(t) is measured in millions What was the rat population in 1992 Preview rats What does the model predict the rat population was in the year 2001 Preview rats



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The population of Delhi, India, can be modeled by $$ P(t)=9817(1.031)^{t} $$ where $P(t)$ is in thousands and $t$ is the number of years since 2001 a. Using this model, estimate the population in the year 2018 . b. In what year did the population reach 15 million ( 15 million is $15,000$ thousand)?

Okay let's say I want to use the exponential growth function of the population of the town. Well that's given by N. F. T. Equals the starting population times E. Um Two the end times T. Where N. Is the amount that it's increasing per year. Okay so what I'm trying to do here is I'm giving some information, I know that the Starting population at T. equals zero Right? The starting amount and is going to be in 1990 and that's starting amount is going to be 22,600. So this will be my starting amount and not. Okay I also know that in 1995 that's five years after starting them out. So I'm plugging in five here fruity Um the population was 24,200. Okay so given this information I'm trying to find my in value I don't know what this is yet. It's the constant. Which which with which this population is going to be growing. So I'm gonna solve for it. Okay I'm going to get the e. By itself. So divide by 22,600. And that's approximately about 1.0707. Approximately This cancels E to the five times end. How do I get rid of the exponential function? Remember take the Ellen. Ellen is the same thing as log base E. So that will cancel an exponential and logarithmic functions are the same. Okay Equals five times in. So I'm gonna divide by five to both sides to figure out what my in value is and plug that in my calculator and that will be approximately Um 0.01 36 eight. Approximately you can around to a couple decimal places. I'm just going around five. So why did I do that? This was just for my original function. I just wanted to find my exponential growth function which is going to be N. F. T. Equals the starting population is still the same. My 22,000 600. He is a constant to the 0.01368 T. Where T represents the years afterwards. Okay Now let's actually apply this. Let's say I wanted to find um using this growth function. The population of the town in 2005. So do I plug 2005 in for tea? No. Remember when T. equals zero? That's 1990s. So how many years after 1990s? This this is 15 years after 1998. -1990. Okay so I'm just gonna plug in 15 into this function that I just found. Which will approximate my amount. So the button is under calculator. And when I get this population will be approximately rounded to the nearest person. 20 Run to the nearest 100 20 7747. How to run to the nearest 100. Right. Look at the Ford has ever end up at seven. no, So it'll be approximately 27,700 people in town in 15 years.

So this question deals with exponential and logarithmic functions, and I'm going to approach it using a natural law of transformation. So for part A, we're just looking for what the population of New York was in the year 2000. So this is our starting year for the equation, which means Time T is just going to be zero. So our equation is just 18.9 times E to the zero, which is just one so a is equal to 18.9. Again, this is measured and million's, so it's 18.9 million now for Part B, we are looking to see when the population is going to reach 19.6 million. So we're gonna plug in 19.6 for A and that's going to be equal to 18.9 times E to the point 00 five five key. And we're gonna want to solve for T the amount of time it takes again. This is measured in years, so we can start by dividing both sides by 18.9. So 19.6, divided by 18.9, gives us 1.37 then that's so equal to R E to the 0.55 t now, because we have a base E, it makes the most sense to take the natural log of both sides of this equation. They're giving us a little more room. You take the natural log of our left hand side natural log of the right, and a natural log of the left hand side gives us about point 0364 And now, on the right hand side, we're taking the natural log something that's based E. So this just basically cancels out the E and you bring down the numerator. So our right hand side is just the point 00 five five teeth. Now we're still solving for tea, So we're going to divide both sides by the 0.55 three of the 30.0 divided back 55 and you should get T is equal to six point 61 cube. So this means that our population is going to reach 19.6 6.612 years after the year 2000, which means it's going to happen sometime during the year 2006. So that's our answer for part B during the year 2006 and again the answer to Part A was just 18.9 million.

So what I'm given in this problem is I'm given the town population at two different times. In 1996 when I played in 1996 towns population is 57,700. And then again in 2000 when I plug in 2000, the town's population is 58,100. Okay so first I want to find a general exponential growth function. What is the general formula for this? Well it could be written in a number of ways. The general formula is whatever the initial population is. Times E. Oilers number two. The R. T. Okay, is there anything I can get from this? Will note that T. Equals zero will represent the population in 1996. So this will actually be my starting population. So my starting population will be 53,700 people. Might end not. Okay. And what else do I need to find? I need to calculate I'm plugging in t into the functions of E. Is already a constant. I don't need to plug it in there. I need to figure out my our value. Okay so he stays saying my our value let's say my time let's say I'm going to use the second point to kind of find time. So t equals zero. Is 1996. So how many years after 1996 is 2004. So if I plug in four I'm going to get out the population of 58. 100. And based on these two points this model I can use this to predict future years. So I'm going to solve for my our value and use that to predict future years. Okay when I divide both sides this will be approximately 1.819 equals E. To the four times are. Okay How do I get rid of the E. Value? What's the inverse of the exponential function? The logarithmic function. And it needs to be the same logarithmic base and note that Ellen is the same thing as log base key. So I could take the Ellen on both sides. Ellen an evil cancel. So I have the Ellen of approximately 1.819 equals four times. Are need to get our by itself. Somebody divide by ford both sides. And when I put this in my calculator, make sure you close the Ellen our is approximately 0.1968 approximately. So what's my population growth? N. F. T. Equals my starting population. Which was 53 70. Um 53,700 E. To the 0.1968 times. T. And then from this I can use this to calculate my further population. Let's say I want to calculate the year 2008. What I plug in 2000 and into my function. No remember it's the years after 1996. So this will be 2000 and eight minus 1996. Which will be approximately 12 years. Let's say I plugged in 12 into this function. I'm just replacing T with 12. Okay so I'm going to raise the exponential function in my calculator. Make sure I multiplied by 12 times my starting value. Okay and when I do this my population should be approximately 68 2000 and four people. And how do they want me to round? They want me around the nearest 100. So the hundreds place is that zero round up that zero, nope. So about 68,000 will be the population in 2000 and eight according to this growth rate.

In this question, the population of China is murdered by the equation. B D equals 1237 multiplied by 1.95 is to the part B. And he's the number of years since 1998. Now, in a part, they're asking, What is the population in the year 2002? It means we have to find the population after four years, so be off. Four equals one toe 37 1.95 days to depart for on. When we calculate this, we get 12 84.680 And after rounding off bigger Oneto 85 million's No Inbee Park, it sees estimated the population in 2016 it means we have to find the population. After 18 years, soapy off 18 equals one toe 37 multiplied by 1.95 days to the part. 18. When we calculate this, we get 1466.5 millions on Finally in C part. It says if the growth rate continues in but here in the population needs two billion people's toe, two billion means 2000 millions. So it means we can equate the equation toe to thousands or 2000 equals. 1237 multiplied by 1.95 raised to the party on Now we have to solve 40. So we start by dividing both sides by 1237 This cancels out on we get 2000 by 1237 equals 1.95 days to the part. Hey, now we have to solve 40. So apply Natural Logan both sides. So Ellen 2000 by 1237 is equal to Ellen. 1.95 Race to the party. So Ellen 2000 divided by 1237 equals D multiplied by Ellen 1.95 So finally, solving for tea, we get these equal to Ellen 2000. Divided by 1237 Holy Waited by Ellen 1.95 Final is holding for TV. Get these equal to 50.8 now. It means we have to wear 50.8 to 1998. So we'll get here. 2040 night


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