This exercise uses the population growth model. The graph...

This exercise uses the population growth model. The graph shows the deer population in a Pennsylvania county between 2010 and 2014. Assume that the population grows exponentially. n(t) A 30,000 Deer population 20,000 10,000 0 1 (4, 31,000) 2 3 4 Years since 2010 (a) What was the deer population in 2010? deer t (b) Find a function that models the deer population t years after 2010. (Enter your answer in the form $n_0e^{rt}$. Round your $r$ value to four decimal places.) n(t) = (c) What is the projected deer population in 2017? (Round your answer to the nearest thousand.) deer (d) Estimate how long it takes the population to reach 300,000. (Round your answer to two decimal places.) yr

Transcript

00:01 Okay, in this problem it says the graph shows the deer population in a pennsylvania county between 2010 and 2014.

00:09 Assume that the population grows exponentially.

00:12 All right, question a says, what was the deer population in 2010? well, 2010 is when the time t is equal to zero.

00:24 So as we look here on the graph, at time zero, the population would be 20 ,000.

00:29 So therefore the population is 20 ,000 in 2010.

00:36 All right, in question b, it says, find the function that models the deer population t years after 2010.

00:44 All right, so for part b, we know the formula is n of t is equal to n sub -zero times e raised to the power of rt.

00:55 That's the format we're looking for.

00:56 We also know that n -sub -zero represents the starting population and that's when t is zero.

01:04 So we already know that is 20 ,000.

01:08 So n -sum -zero is going to be 20 ,000.

01:11 Now take a look at this point here.

01:13 That's 4 ,000, 3 ,000.

01:15 That means when t is 4, or in other words, four years later, the population grew to 31 ,000.

01:23 So we can say that when t is 4, the n of 4, which is the value, is 31 ,000.

01:32 So putting all of that together in an equation, we can say that n -0, which is 20 ,000 times e raised to the r, that's r times t, we can substitute t with four, has to equal the 31 ,000.

01:53 Next, we can simply solve for r using algebra.

01:57 First, we'll divide both sides by 20 ,000, and that gives us e to the 4r is 31 ,000.

02:04 Divided by 20 ,000, which reduces to 31 over 20.

02:10 Now, to solve for r, we would have to take the natural log of both sides.

02:16 So the natural log of e to the 4r has to equal the natural log of 31 over 20.

02:25 Now, when you do the natural log of e raised to a power, it's just equal to the power...

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