A population of rabbits grows exponentially the colony was...

A population of rabbits grows exponentially. The colony was first counted in 2010. In 2021 there were 40 rabbits, and in 2015 there were 20 rabbits. a) Give a formula of the form R = f(t)= ab^t for the population of the colony of rabbits as a function of time, where t = the numbers of years after 2010. Identify exact values for both a and b and once this is done, go ahead and round these values to the nearest thousandth. b) Use a graph to determine how long it takes the rabbit population to reach 1000 rabbits.

Transcript

00:01 In the given question we are told that there is a population of rabbits that are growing exponentially and the colony was first counted in 2010.

00:11 Now we are told that in 2021 the number of rabbits were 40 and in 2015 the number of rabbits were 20 in this colony.

00:21 Now we are told to find a formula of the form f of t is equal to a times b to the power of t where f of t would give us the population of rabbits after t years, t years since the first count which is in 2010 and t over here would be the time that is taken in years.

00:59 So, the information that we have is since the years are counted from 2010, when t is equal to zero, it is the year 2010, right? so when t is equal to 5, it should be the year 2015, and when t is equal to 10, the year should be 20 -20 so when t is equal to 11 the year would be 20 -21 so now what we can do is we can write when f of t when f of zero is taken a times b to the power of zero is what we would have which is equal to a so f of zero is equal to a is what we have got over here next we can can take f of 5 f of 5 is the population in 2015 which is a times b to the power of 5 which we are told to be equal to 20 and f of 11 which is in the year 2021 a times b to the power of 11 is equal to 40 right so let's call this equation number 1 and let's call this equation number 2 now if we were to equation number 2 divided by equation number 1 what we would have is a times b to the power 11 divided by a times b to the power of 5 is equal to 40 by 20 so a divides a b raised to the power 11 divided by b raised to the power 5 simplifies to be raised to the power 11 minus 5 40 by 20 is equal to 2 11 minus 5 is equal to 6 so b raised to the power 6 is equal to 2 or we can write b is equal to 2 raised to the power 1 by 6 so now when we evaluate this using a calculator when we take the power of 2 as 1 by 6 what we would have as the value over here is 1 point value of b is 1 .1 .1 .1.

03:27 225.

03:28 So now that we have the value of b we can substitute it in equation number let's say 2, let's substitute it in equation number 1.

03:39 So we will have a times 1 .1 raised to the power 5 is equal to 20 which means a is equal to 20 divided by 1 .1225 raised to the power of 5.

03:58 So on evaluating this, we would get the answer over here to be equal to the value of a, what we get over here is 11 .11 .225.

04:18 But what we have found already over here is that a is equal to f of 0, which means a is is the initial rabbit population initial rabbit population and since it is a population we should get the answer in a whole number right so we can approximate 11 .225 as 11 rabbits so a is equal to since it is more than 11 let's approximate it to 12 right so a is equal to 12.

04:55 So a is equal to 12 rabbits.

04:58 So now that we have the values of a and b, we can write f of t as equal to 12 times 1 .1 225 raised to the power of 5, raised to the power of t is the model that uses the population of the rabbits...

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