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In the given question we are told that a squirrel population was introduced in a region 21 years ago and we are told that biologists observe that the population doubles every 7 years and now the population is 680.
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First we are told to find the initial squirrel population of this city, of this region.
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So what we do know is that the the skeural population was introduced, introduced 21 years ago, right? it was introduced 21 years ago.
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And we are also told that it doubles every seven years, doubles every seven years.
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And also that the current population is 680 the current population is 680 the current population is 680 so now to find the initial squirrel population what we can first do is we can form an exponential equation which is of the form let's call it p of t equals p zero times e to the power of k times t and in this exponential equation we can define each terms so p of t is the population after population after uh t years t years p zero is the initial population is the initial population and k over here is a constant and t is the time taken in years time taken in years so what we do know is that when t is equal to zero it is giving us the year which is 21 years ago right so it is given giving us the year which was 20 years ago right so it is given giving us the year which was 21 years ago when the squirrels were introduced and in that time p of 0 that is the population of squirrel would be p 0 times 8 to the power of k times 0 that is p equal to 0 over here and what we would have is p 0 right so this p 0 is the initial squirrel population now what we are told is that after 7 years this population doubles which means when we take p of 7 it is equal to p 0 times e to the power of k times 7 and this is actually 2 times p 0 right after 7 years the population is the double of the initial population so what we can do over here now is we can write we can take this equation over here and on dividing each side with b0, we will have e to the power of 7k is equal to 2, right? and in the next step, what we do is we take the natural log on each side of this equation and the reason why we are taking the natural log is to use a property of natural log that says when we take the natural log of e raised to any power, let's say c, the value of this expression is simply equal to the power of e.
03:52
So over here the power of e is 7k so the value of this expression would reduce to 7k according to this property and 7k would be equal to the natural log of 2.
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So from this we can write k is then equal to the natural log of 2 divided by 7.
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So when we take the natural log of 7 and divide it when we take the natural log of 2 and divide it with 7 what we get as 7 and divide it, when we take the natural log of 2 and divide it with 7, what we get as 7.
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The value of k is 0 .099.
04:27
So now that we have figured out the value of k, we can write that what we know in this question is that the third condition is that after 21 years, that is today, the population is 680 square else, right? so this is the population today...