Use the exponential growth model, A=A0 e^k t, to show that...

Use the exponential growth model, $A=A_{0} e^{k t},$ to show that the time it takes a population to triple (to grow from $\left.A_{0} \text { to } 3 A_{0}\right)$ is given by $t=\frac{\ln 3}{k}$

Transcript

00:01 For this problem, we'll be using the exponential growth model, a equals a sub 0, or a not, a to kt, to show that the time it takes a population to triple, it's given by t is equal to natural log of 3 divided by k.

00:18 Okay, so let's focus on this exponential growth model.

00:22 A is equal to a sub 0 or a not, e to kt.

00:28 So the different parts to it we have here, just to do a little bit of review, is that the a sub -zero part, that's the initial population.

00:42 A by itself is going to be a future population or any population after what you had started with.

00:51 In particular, this future population is at time t, and then k is the rate.

01:01 So the rate at which this population is growing.

01:05 So in particular, we're interested in a population tripling.

01:09 So three times the initial population.

01:13 So to throw that in there, three times the initial population will look like this.

01:18 It will look like 3a sub 0 on the left hand side of the equal sign.

01:24 And in everything else on the right hand side, we can keep it the same.

01:29 Since we're answering this generically, i'm not going to throw in any other numbers.

01:35 And basically what we need to do is we need to solve for t.

01:40 So find the time at which the population is going to be triple of what it was.

01:46 So solve for t.

01:47 Okay, so let's get started doing that.

01:50 To solve for t, first thing i'm going to do is i'm going to divide both sides by a sub -zero.

01:56 Since we have them on both sides...

Question

Use the exponential growth model, $A=A_{0} e^{k t},$ to show that the time it takes a population to triple (to grow from $\left.A_{0} \text { to } 3 A_{0}\right)$ is given by $t=\frac{\ln 3}{k}$

Please give Ace some feedback