Question

Let P(t) be the population of a country, in millions, t years after 1990, with P(6) = 3.21 and P(11) = 3.69.

Find a formula for P(t) assuming that it is exponential.

Answer 1

@zepdrix

Answer 2

What type of class is this for? :)
I don't want to start with it in differential equation form if you're taking algebra or something similar ^^ heh

Answer 3

Its algebra. Im pretty sure im supposed to use \[y=c*b^x\]

Answer 4

I have to figure out c and b with the two points given

Answer 5

Ok let's use the information given to FIRST find b.
It's going to be a little bit confusing since we'll do a little bit more than simply add or subtract equations. :D
Let's first start with our equation like this, since it's in terms of Population and time.\[\huge P(t)=Cb^t\]

Answer 6

\[\large P(6)=3.21 \qquad \rightarrow \qquad 3.21=Cb^{6}\]\[\large P(11)=3.69 \qquad \rightarrow \qquad 3.69=Cb^{11}\]Understand how I got these two equations? Plugging 6 and 11 into our model.

Answer 7

Affirmative

Answer 8

From here we can DIVIDE these equations to ELIMINATE the C term. Which will allow us to solve for b.\[\large \frac{P(6)}{P(11)} \qquad \rightarrow \qquad \frac{3.21}{3.69}=\frac{Cb^6}{Cb^{11}}\]I probably should have divided them the other way, with the 11 on top :) But oh well, it'll still work out.

Answer 9

I didnt know you could do that =p

so it gives

3.21/3.69=b^-5?

Answer 10

yes, from there let's change it to a +5 then take the 5th root of both sides.

Answer 11

I don't just mean, "change it", I mean find the equation in terms of b^5, heh. If that part is confusing, we can do the steps :D

Answer 12

Hmmm im not sure...would i change it to like 1/b^5?

Answer 13

Yes, then you can rewrite BOTH sides as reciprocals :)

Answer 14

If that is confusing, you can always just cross multiply or something similar ^^

Answer 15

yeah not sure what ya mean there.

3.69=3.21b^5
3.69/3.21=b^5

\[\sqrt[5]{3.69/3.21}\]

did i do that right?

Answer 16

Yah looks good :)
Now the only problem we'll have from here is....
If we want to accurately get C, we'll want to leave b like that for now.
We don't want to round at this point, since we still need to solve for another variable.

Answer 17

From this point, it shouldn't be too bad.
Let's plug our b value into P(6) or P(11) and solve for C.

Answer 18

C represents the INITIAL POPULATION. So hopefully we'll get a number that makes sense :o

Answer 19

Hopefully.

\[3.21=c*\sqrt[5]{3.69/3.21}^{6}\]

Answer 20

that looks funky

Answer 21

\[\large P(6)=3.21 \qquad \rightarrow \qquad 3.21=C\left(\sqrt[5]{\frac{3.69}{3.21}}\right)^6\]Hehe it sure does :3 you can type "large" or "huge" (without the quotes) in your equation tool to make it a little larger if you need :D cause the exponents can be difficult to read sometimes.

Answer 22

Didnt know that either =p

Do you subtract the fifth root and the exponent 6?

Answer 23

Hmm this is giving us a really bad number :( Are you sure we used the correct model? I'm more familiar with seeing this model for population.\[\huge P(t)=Ce^{kt}\]But if that's not one that you've been using, then ignore it.

Answer 24

oopsy. :(

Answer 25

How do i know which one to use? i assumed it was the first one because thats what we learned in class today, and this problem is on the homework

Answer 26

But we used the one you just typed a few days ago

Answer 27

And i thought c was initial population but the problem doesnt say the initial population

Answer 28

Wait...i used decimals for b and c and typed the final function into my homework and it said it was correct. So we did it right :) well mostly you lol. Thank you

Answer 29

Oh weird! :) Yes C should represent our initial population. But it's giving me a value of like 2.71... So the oh oh oh oh... it's in MILLIONS.

So the initial population was 2.7 million.
Ok that makes A LOT more sense! :) I guess we were doing it correctly.
I'm sorry about the slow response.
I was trying to work on multiple problems at once, and the site is lagging reallyyy bad right now :( hard to move back and forth.

Answer 30

It's okay. Makes tons more sense. Thanks again =D