Question

Find the annual growth rate if a population has grown exponentially from 40 million to 50 million in the last 20 years. At this rate, what will be the population 10 years from now?

Answer 1

Okay, what can you do on your own?

Answer 2

We have the formula: \[
a(t) = a_0r^t
\]

Answer 3

We can let \(a_0=50\) in millions, and say that \(a(-10) = 40\).
Then we just want \(a(20)\).

Answer 4

\[
40 = 50r^{-10}
\]Can you solve for \(r\)?

Answer 5

how come it's a negative 10?

5/4r^-10?

Answer 6

We are saying it was initially at 50, and 10 years in the past it was 40.

Answer 7

oh okay i understand

Answer 8

to solve for r do i just divide 50 to both sides?

Answer 9

That's a start.

Answer 10

5/4?

Answer 11

No, not quite.

Answer 12

can you help go through it then?

Answer 13

?

Answer 14

@wio

Answer 15

\[
\frac{4}{5} =r^{-10} \\
\frac54=r^{10}\\
r = \sqrt[10]{\frac 54}
\]

Answer 16

Actually, I have a better idea. Let's just say that \(t\) is for every \(10\) years, instead of for every \(1\) year.

Answer 17

If we do that, then \[
r = \frac 54
\]

Answer 18

And we just want \(a(2)\) where: \[
a(t) = 50\left(\frac 54\right)^t
\]

Answer 19

so…

2=50(5/4)^t

?

Answer 20

No, not \(a(t)=2\). I said \(a(2)\). \(t=2\).

Answer 21

oh so just 50(5/4)^2 ?

Answer 22

Yes.

Answer 23

Wait, I am wrong..

Answer 24

whats wrong

Answer 25

It should be \(a(1/2)\).

Answer 26

I said that \(t\) will be every ten years. But it really was every 20 years. So ten years from now is \(1/2\)

Answer 27

55.90 for the answer?

Answer 28

and how did you get 1/2?

Answer 29

@wio