How Is the Growth Constant Derived in Exponential Population Models?

[supernova1203]

Homework Statement

In the 2 following problems they use the term in the brackets differently, in one case its a percentage and in the other case i have no idea where they get the number from, this is what i would like to find out

A cell loses 2% of its charge every day
C is total charge t is time(measured in days)

C(t)=100(0.98)^t

so basically for each day we put a number in the exponent, for example if 4 days have passed then we write

C(t)=100(0.98)⁴

So here 100 represents the total charge, 0.98 represents the percentage of charge left after 1 day. This is one type of exponential growth/decay problem I see and is easy to solve.

This other type of problem is where i am having a bit of trouble:

The population of a country in 1981 was 24 million

P represents the population in millions and t represents the time in years

The following equation represents this model

P(t)=24(1.014)^t
Here 24 represents the total population in 1981, 1.014 is the growth rate by which the population increases each year(in the millions) and the exponent is where we put in the number of years it has been since 1981

so for example if we use 2011 (1981+30=2011) that means to get the population for 2011
we do P(t)=24(1.014)³⁰

which gives us the answer

P(t)= 24(1.517534768)
P(t)=36.4 million.

my question is where or how do they get the 1.014 constant which is used to calculate the population for every year relative to 1981? It is the exponential growth multiple isn't it? How did they get it?

Homework Equations

P(t)=24(1.014)^t

The Attempt at a Solution

 

[eumyang]

I usually think of exponential growth/decay models in terms of this equation:
$$P(t) = P_0 (1 + r)^t$$
r is the rate of growth/decay. r > 0 would indicate a growth, and r < 0 would indicate a decay.

So for the 1st equation,
$$C(t) = 100(0.98)^t = 100(1 - 0.02)^t$$
... since a cell loses 2% of its charge every day.

For the 2nd equation,
$$P(t) = 24(1.014)^t = 24(1 + 0.014)^t$$
... so the rate of growth is 1.4% per year since 1981. This information should have been given in the problem somewhere.

 

[supernova1203]

I usually think of exponential growth/decay models in terms of this equation:
$$P(t) = P_0 (1 + r)^t$$
r is the rate of growth/decay. r > 0 would indicate a growth, and r < 0 would indicate a decay.

So for the 1st equation,
$$C(t) = 100(0.98)^t = 100(1 - 0.02)^t$$
... since a cell loses 2% of its charge every day.

For the 2nd equation,
$$P(t) = 24(1.014)^t = 24(1 + 0.014)^t$$
... so the rate of growth is 1.4% per year since 1981. This information should have been given in the problem somewhere.

so the only difference between the first and the second equation is that the first equation rate(r) is not multiplied by 100 and the second one rate(r) is?

 

[eumyang]

so the only difference between the first and the second equation is that the first equation rate(r) is not multiplied by 100 and the second one rate(r) is?

I don't know what you mean. r = -0.02 corresponds to a 2% loss rate. r = +0.014 corresponds to a 1.4% rate of growth. We're just converting to percents, and in both cases we multiply 100 to convert to percents.