Exponential Growth- Am I doing this right?

[Lo.Lee.Ta.]

1. Let's say that at the end of 2010, the population of Athens was 100,000. Let's also say that the relative growth rate of the population was 2% per year.
What's the population at t = 0.5 years after the end of 2010?

2. To solve this sort of problem, I thought I'd have to use: y= A(1+b)^t

A is the initial population

b is the growth factor

t is the timeI thought that since the population growth rate was 2% per year, the growth factor would have to be (.001 + 1).

I figured out the .001 by saying: (.2%/yr) = (x/.5yr)
x= .1% or .001
(.1%/100%) = .001 = b

y= 100,000(1 + .001)^.5

y= (100,000)(1.001)

y= 100,100 people

Is this right?
Thanks for the help! :)

 

[Mark44]

1. Let's say that at the end of 2010, the population of Athens was 100,000. Let's also say that the relative growth rate of the population was 2% per year.
What's the population at t = 0.5 years after the end of 2010?

2. To solve this sort of problem, I thought I'd have to use: y= A(1+b)^t

A is the initial population

b is the growth factor

t is the time


I thought that since the population growth rate was 2% per year, the growth factor would have to be (.001 + 1).

I figured out the .001 by saying: (.2%/yr) = (x/.5yr)

You're mixing up percentages and decimal numbers. .2% is a small fraction of 1%.

x= .1% or .001
(.1%/100%) = .001 = b

y= 100,000(1 + .001)^.5

y= (100,000)(1.001)

y= 100,100 people

Is this right?
Thanks for the help! :)

 

[haruspex]

As a check, try calculating the population after 1 year. Does the result seem reasonable in relation to your answer for 6 months? (Hint: it isn't.)
But there's a subtler problem. Over such a timescale, population growth is approximately a continuous process. That is, in a small time δt, the population will increase by a factor 1+λδt. That gives you a differential equation. Solve that, then use the annual growth rate to determine λ.

 

[Lo.Lee.Ta.]

What a minute. Wouldn't my whole (.2%/1yr)=(x /100%) be unnecessary in the first place?

Wouldn't I need to do (2%/100%) = .02

Shouldn't it instead be: y= 100,000(1 + .02)^.5 ?

y= (100,000)(1.00995)

y= 100,995.0

 

[haruspex]

What a minute. Wouldn't my whole (.2%/1yr)=(x /100%) be unnecessary in the first place?

Wouldn't I need to do (2%/100%) = .02

Shouldn't it instead be: y= 100,000(1 + .02)^.5 ?

y= (100,000)(1.00995)

y= 100,995.0

Yes, that's fine.

 

[Lo.Lee.Ta.]

Oh, okay. Thank you! :D