Question

help me with this I am stuck please help me finish I just need an explanation please. the population of a city is growing at a rate proportional to its population. the population 20 years ago was 100 000 and today it is 150 000. find the population 20 years from now

Answer 1

I need to apply logarithms application to this problem.

Answer 2

From a calculus perspective, you can start with \[dP = \alpha Pdt\] where P(t) is the population This will give you something like \[P(t) = Ce^{\alpha t}\] with C and a being constants.

Consider using "20 years ago" as t=T. Then "today" is t=T+20. Plugging these in you have two equations:\[100,000=Ce^{\alpha T}\]\[150,000 = Ce^{\alpha(T+20)}\]
If you divide the second equation by the first, you get:
\[1.5 = e^{\alpha 20}\] and you can find a. Then, do something similar with "twenty years from now" (t=T+40!)...
\[100,000 = Ce^{\alpha T}\]\[X = Ce^{\alpha (T+40)}\]
Divide the second by the first:
\[\frac{X}{100,000} = e^{\alpha 40}\]
Where you know a! Solve for your answer X.

Answer 3

can you help please

Answer 4

Which part doesn't make sense?

Answer 5

150 000=ce^at

Answer 6

I have never seen that can you use the general form of A=Poe^kt

Answer 7

You have seen \[A = P_0e^{kt}\] right? So we re-word it a bit. :) But let's call "twenty years ago" t = 0, okay?

Answer 8

yes

Answer 9

Then you can solve for P_0

Answer 10

So what is P_0?

Answer 11

Po=100 000

Answer 12

Okay, so now we have \[A = (100,000)e^{kt}\]

If twenty years ago was t=0, what does t equal today?

Answer 13

is it A_o or A

Answer 14

One step at at a time. t is the time in years, right? So, t=? today?

Answer 15

20

Answer 16

Okay, so now we have (for today)
\[A = (100,000)e^{20a}\]
AND we know that the population today is 150,000. Where does population go into that equation?

Answer 17

goe to A

Answer 18

Great. So you have \[150,000 = 100,000e^{20a}\] And you can solve for a. So now, find a.

Answer 19

ok doing it now just a sec

Answer 20

got 0.0202

Answer 21

Okay. Good. Now you know a (or k) and P_0. So your population equation is:\[A = (100,000)e^{0.0202t}\]

Finally, twenty years from now, t=?

Answer 22

will it be 20 again?

Answer 23

Not quite. t=0 was twenty years ago. t=20 is today. t=? is in another twenty years, so t=?

Answer 24

40

Answer 25

Yep. So plug in t=40, and find A.

Answer 26

I got 100 000e^0.808

Answer 27

Good. Keep going

Answer 28

(Plug in e so you get the final answer)

Answer 29

I did and the final result is 14.2486...

Answer 30

Hmm, that sounds a little low. What is e^0.8?

Answer 31

yes did to me too

Answer 32

I have been making mistakes when I get to this step

Answer 33

Are you using a calculator on the computer or a handheld calculator?

Answer 34

handheld

Answer 35

Which model?

Answer 36

(Once you get the gist of it, you can practice, and check your answers using Google. If you type e^0.8 into Google, you will get the right answer!)

Answer 37

casio scientific calculator

Answer 38

i dont know if i need to divide or take log of natural number

Answer 39

For e^0.8, you are raising the number e (~2.7...) to the 0.8 power.
If you calculator looks like this:
http://www.ivgstores.com/prodimages-cdls/pet-casi/CIOFX115MS-L.jpg
then try pressing "shift" then "ln" then 0.8.

Answer 40

ln 0.8/ln 100 000 ?

Answer 41

ln is the opposite of e. Think of it like ln takes the square root of a number, and e squares a number.

Answer 42

so sorry I dont understand what you mean

Answer 43

Let's just say that you have a number 2.
You can multiply 2 by 4. You can divide 2 by 4. Those do the opposite things, right? Well, logs (ln) and exponentials (e) also do the opposite things.

Answer 44

In this way:
\[ln(2) = 0.69\] and \[e^{0.69} = 2\]

Answer 45

And from this, you get properties like \[\ln{(e^x)} = x\] and \[e^{\ln(x)} = x\] (Which is like saying "If I do multiplication by 4 and division by 4 to the number 2, I get the number 2 back!)

Answer 46

I still get a lower number

Answer 47

Try it in Google first. What does google say is e^0.8?

Answer 48

=2.24341666

Answer 49

Okay, so then \[ 100,000e^{0.8} = 224,342\] Which is reasonable. Now you just need to figure out how to get 2.2 in your calculator for e^0.8

Answer 50

I did shift e^0.808

Answer 51

Did the display in your calculator say e^0.808? If so, then you should get 2.24 in your caculator

Answer 52

yes

Answer 53

Good. Now do you have your answer?

Answer 54

wow what a day. i just have to do 100 000e^0.08

Answer 55

Well, not e^0.08! Make sure you do 100 000e^0.808

Answer 56

oh yeah I got 224.3416664

Answer 57

224,312 people sounds reasonable if today there are 150,000 and twenty years ago there were 100,000 people.

Answer 58

yes. your great thanks for your patience and your help

Answer 59

no problem. keep working with your calculator. keep practicing. it will get more natural!

Answer 60

I will