Complete table math

Question

Complete the following table 

 

Population    Growth Rate,k    Doubling Time ,T

Country A     2.4% per year

Country B                              63 years

 

 

Country A doubling time is?

Country B % per year is?

Answer 1

Hi Logan,

 

We want to use the population growth equation

 

P(t) = P₀r^t

 

where

 

P(t) = current population

P0 = initial population, or population at time t = 0

r is the growth rate

 

For Country A, we are told that the population is growing at 2.4% per year. So, r = 1.024. The doubling time is when

P(t) / P₀ = 2.

 

We can write the growth equation as

 

r^t = P(t) / P0 = 2

 

r^t = 2

 

Take the natural log of each side.

 

ln(r^t) = ln(2)

 

Use the law of logarithms I spoke about in my other answers to write t as a coefficient.

 

t ln(r) = ln(2)

 

t = ln(2)/ln(r)

 

r = 1.024

 

So, t = ln(2)/ln(1.024)

 

t = 29 years This is the doubling time in the first row of the table

 

In the second row of the table, we are told that the doubling time for Country B is 63 years. This means that at t = 63 years, P(t) / P₀ = 2.

 

Again, we can write the original exponential growth equations as

 

r^t = P(t) / P₀ = 2

Take the natural log of each

 

ln(r^t) = ln(2)

 

In the same way, we can write t as a coefficient

 

tln(r) = ln(2)

 

Then

 

ln(r) = (1/t)ln(2)

 

We can then exponentiate each side, remembering that e to the natural log of an argument is just the argument.

 

r = e^(1/t)ln(2)

 

or

 

r = e^(1/63)ln(2)

 

So, r = 1.011

 

This means that Country B is growing at 1.1%

 

Your table should look like

 

 

 Population    Growth Rate,k    Doubling Time ,T
Country A     2.4% per year      29 years
Country B     1.1% per year       63 years