Answers to growth rate exercises:

Sociology 329

1. United States

a. Average annual rate of growth from 1950-1960
P₁ = 151,326
P₂ = 179,323
(population in thousands)
t = 10
r = ?

r = (ln [p2/p1]) / t
r = (ln [179323/151326]) / 10
r = (ln[1.185]) / 10
r = (0.1697) / 10
r = 0.01697 --> 1.697% average annual growth

b. Doubling time:

    t = ln(2)/r
    t = .693/0.01697
    t = 40.835 years

c. Estimated Population in 2010
P₁₉₆₀ = 179323
r = .01697
t = 50
P₂₀₁₀ = ?

P₂ = P₁ * e^rt
P₂ = (179323)* e ^{(0.01697)(50)}
P₂ = (179323)* e^(0.8485)
P₂ = (179323) * (2.33614)
P₂₀₀₆ = 418,924 (in thousands)

2. Rural United States

a. Average annual rate of growth from 1950 - 1960
P1 = 54,479
P2 = 54,054
(population in thousands)
t = 10
r = ?

r = (ln [p2/p1]) / t
r = (ln [54054/54479]) / 10
r = (ln[0.9922]) / 10
r = (-0.0078) / 10
r = -0.00078 --> -.078% average annual growth

b. Doubling time is undefined when growth rate is negative. (If the population is shrinking every year, it can never double)

But, you can calculate the halving time, using same formula (just remember to drop the negative sign):

T = (.693)/-.00078 = 888.46 years for the rural population to be cut in half, at current rate of growth.

c. Estimated Rural Population in 2010
P₁₉₆₀ = 54,054 (in thousands)
r = -0.00078
t = 50
P₂₀₁₀ = ?

P₂ = P₁ * e^rt
P₂ = (54054)* e ^{(-0.00078.)(50)}
P₂ = (54054)* e^(-0.039)
P₂ = (54054) * (0.96175)
P₂₀₁₀ = 51,986 (in thousands)