Exponential growth formula

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Mrtinkles

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Hello, I am wondering if anyone can help me with this?

The global human population was approximately 1.6x109 in 1900, and 6.1x109 in 2000. The population dynamics can be modelled by the exponential growth formula Nt = N0e kt.

Using the above information, write a general expression for the rate constant k and then evaluate this expression to evaluate this constant to 3 significant figures (Use years as the unit of time).
 
It's very difficult to know where you need help. See the "read before posting" rules specifically the section, "Show some of your work or explain where you're stuck".

You've been given two sets of data, one for 1900 and one for 2000. Can you substitute this data into the given formula to obtain two equations?
 
Mrtinkles said:
Hello, I am wondering if anyone can help me with this?

The global human population was approximately 1.6x109 in 1900, and 6.1x109 in 2000. The population dynamics can be modelled by the exponential growth formula Nt = N0e kt.

Using the above information, write a general expression for the rate constant k and then evaluate this expression to evaluate this constant to 3 significant figures (Use years as the unit of time).
Click to expand...
Because t hasn't been defined, apart from its unit, you are free to decide what year to take as t=0. Commonly we would define t as "years since 1900", to keep the numbers small. But you could also define t as the year number itself. This will not affect the answer to the question, though the expression will be simpler in the former case.
 
Also, if you are not using "Latex" please indicate powers by "^". That is, you mean 1.6 x 10^9 not 1.6 times one hundred nine!

You are told that the formula is \(\displaystyle N_t= N_0e^{kt}\). Taking 1900 to be t= 0, saying that the population in 1900 was 1.6 x 10^9 means that \(\displaystyle 1.5 x 10^9= N_0e^{k(0)}= N_0\) so \(\displaystyle N_0= 1.5 x 10^9\).

Saying that the population in 2000 (t= 2000-1900= 100) is 6.1 x 10^9 means that \(\displaystyle 6.5 x 10^9= N_0e^{k(100)}= 1.5 x 10^9e^{100k}\).
\(\displaystyle e^{100k}= \frac{6.5 x 10^9}{1.5 x 10^9}= \frac{65}{15}= \frac{13}{3}\).

\(\displaystyle 100k= ln\left(\frac{13}{3}\right)\)
\(\displaystyle k= \frac{ln\left(\frac{13}{3}\right)}{100}\).
 
HallsofIvy said:
Also, if you are not using "Latex" please indicate powers by "^". That is, you mean 1.6 x 10^9 not 1.6 times one hundred nine!

You are told that the formula is \(\displaystyle N_t= N_0e^{kt}\). Taking 1900 to be t= 0, saying that the population in 1900 was 1.6 x 10^9 means that \(\displaystyle 1.5 x 10^9= N_0e^{k(0)}= N_0\) so \(\displaystyle N_0= 1.5 x 10^9\).

Saying that the population in 2000 (t= 2000-1900= 100) is 6.1 x 10^9 means that \(\displaystyle 6.5 x 10^9= N_0e^{k(100)}= 1.5 x 10^9e^{100k}\).
\(\displaystyle e^{100k}= \frac{6.5 x 10^9}{1.5 x 10^9}= \frac{65}{15}= \frac{13}{3}\).

\(\displaystyle 100k= ln\left(\frac{13}{3}\right)\)
\(\displaystyle k= \frac{ln\left(\frac{13}{3}\right)}{100}\).
Click to expand...

Thanks for the replies, sorry I am terrible at writing it up. I will keep it in mind. I'm very bad at maths so hopefully you can help some more.
I am wondering how you got 1.5 x 10^9? and how 1.5 x 10^9= N_0e^{k(0)} can = N_0? Is it because e^{k(0)} = 0 so can just be removed?
 
Mrtinkles said:
Thanks for the replies, sorry I am terrible at writing it up. I will keep it in mind. I'm very bad at maths so hopefully you can help some more.
I am wondering how you got 1.5 x 10^9? and how 1.5 x 10^9= N_0e^{k(0)} can = N_0? Is it because e^{k(0)} = 0 so can just be removed?
Click to expand...
No!

It is because:

e^0 = 1

multiplying by 1 does not change the number.
 
Okay, so where does it go to become N0?
And why does 1.6 x 10^9 become 1.5 x 10^9?
 
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To understand the language of math, you need to pay close attention to how words and symbols are defined.

In the general formula

[MATH]N_t = N_0 * e^{kt}[/MATH]
Nt stands for number (N) after t units of time since the beginning. So what does N0 stand for?

Things make more sense if you know what they mean exactly. Vague will not do it. And note that the following makes sense:

[MATH]\text {If } t = 0, \text { then } N_0 * e^{kt} = N_0 * e^{k * 0} = N_0 * e^0 = N_0 *1 = N_0.[/MATH]
OK

What then is N0 numerically in this problem? What is t numerically? What is Nt?

So you can restate the general formula as a single equation with one unknown.

What kind of equation is it, do you know?

How do you solve that type of equation?
 
Oh I see, that makes sence to me now. I am only confused why 1.6 x 10^9 become 1.5 x 10^9 in this?
 
I have another few parts to the question if anyone can help

The global human population was approximately 1.6x10^9 in 1900, and 6.1x10^9 in 2000. The population dynamics can be modelled by the exponential growth formula Nt = N0e kt. :

b) Calculate the percentage by which the population grows each decade.

So I think... 6.1 / 2 decades = 3.05x10^9 per decade. C=x1 - x2 / x1 = 3.05x10^9 - 1.6x10^9 / 1.6x10^9 = 0.90625 x 100 = 90.625%

I'm not sure what I should do with these two?

c) Assuming that this rate of exponential growth continues, when will the global population reach 1010?


d) Assuming that this rate of exponential growth also applied before 1900, estimate what the global population was in the year 1800, to an appropriate number of significant figures
 
You never answered my questions in post 10. Show us how you got part a.
 
Okay does this sound right?

t = the time = 0
but I don't understand why it's not 2000-1900 = 100
N0 = starting time = 1.5 x 10^9 in 1900
Nt = time after t = 6.1 x 10^9 in 2000

But the post 4th post had these the other way around why is it done that way?
 
You were told that "Nt = N0e kt". Didn't the problem say anything about what "t" and "Nt" represented? You seem to be thinking that the "1.6x10^9" and "6.1x10^9" are "t". No, t is "time" while N is "population". Then you are also free to choose what "t" represents the time. Since you are told values at "1900" and "2000" you can take t to be 1900 and 2000. Then N0e^(1900k)= 1. 6x 10^9 and N0e^(2000k)= 6.1x 10^9. That is two equations to solve for N0 and k.

But it might be simpler to designate 1900 as t= 0 and 2000 as t= 100 so that "t" is "number of years since 1900". Then you need to solve N0e^(0k)= N0= 1.6 x 10^9 and N0e^(100k)= 6.1x 10^9.

Or you can take t= 0 and t= 10 so that "t" is number of decades since 1900. Then you need to solve N0= 1.6x 10^9 and Noe^(10k)= 6.1 x 10^9.

Those will give different values for N0 and k so it is necessary to state what "t" represents!
In any
 
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