Sociology 329

Answers to growth rate exercises

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 2006
P₁ = 179323
r = .01697
t = 46
P₂₀₀₆ = ?

P₂ = P₁ * e^rt
P₂ = (179323)* e ^{(0.01697)(46)}
P₂ = (179323)* e ^(0.78062)
P₂ = (179323) * (2.1828)
P₂₀₀₆ = 391,431 (in thousands)

** compare this to 298,398,484, the official July 1, 2006 population estimate for the U.S from the Census Bureau.  Our growth rate has slowed significantly since the 1950s, the height of the Baby Boom.

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 2006
P₁ = 54,054 (in thousands)
r = -0.00078
t = 46
P₂ = ?

P₂ = P₁ * e^rt
P₂ = (54054)* e ^{(-0.00078.)(46)}
P₂ = (54054)* e ^(-0.03588)
P₂ = (54054) * (0.96476)
P₂₀₀₆ = 52,149 (in thousands) . This can be compared to the 2004 rural population estimate of 59,061,367. (United Nations, Demographic Yearbook).

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