
Prince William County, Virginia which is located 30 km south of Washington D.C., is one of the fastest growing entities in that region. Between 1940 and 1990, the population growth rate was 4% per year.
Population Growth
When population grows exponentially, the amount of time it takes for the population to double in size is termed the doubling time.
Calculating Doubling Time
T = (ln 2/growth rate) x 100   T = doubling time
ln 2 = 0.693
growth rate is the percentage increase or decrease

Multiplying both sides by 100 gives
EXAMPLE 
How long will it take to double the population if the rate of population growth is 7%?
T = 69.3/growth rate
Rounding 69.3 to 70 allows us to simplify the mathematics.
T = 70/7 = 10
A population growing at 7% will double about every 10 years.
After 20 years, the population will be four times as large as it originally was. 
Doubling time is useful for projecting and analyzing the implications of growth (i.e.,the rate of increase of electricity consuption, rate of farmland loss, deforestation rate, the depletion of natural resources such as coal, oil, or natural gas, etc).
EXAMPLE OF THE IMPACT OF EXPONENTIAL GROWTH 
"Long ago in India, according to legend, the inventor of the game of chess so delighted his king that the king asked him to choose one of a number of 'princely' rewards. The inventor bowed and modestly declined, murmuring that he would be happy if the monarch would allow him some wheat to feed his family. He asked that the monarch place one grain of wheat on the first square on his chessboard, two on the second, and double the award for each additional square. The monarch smiled and eager complied, calling for his steward to bring the grain."^{*}
A chessboard consists of 64 squares. How many grains of wheat did the inventor request?

^{*}Bartlett, A.A. Forgotten fundamentals of the energy crisis. American Journal of Physics 1978, 46, 876888.

PROBLEMS 
 At a growth rate of 4%, what is the doubling time for the population of Prince William County, Virginia?
 According to 1990 U.S. Census data, the population of Prince William County
was 215,000. Using the equation t = (1/k)ln(N/N_{0}) and the
doubling time formula, determine when the population of Prince William
County will reach 1 million.
 The area of Prince William County is approximately 9 x 10^{8} m^{2}.
At a 4% growth rate, when would Prince William County reach a population
density of 1 person per square meter?
 What would be the population of Prince William County after 10 doublingtime intervals?

Impervious Surface
Development, the erection of humanbuilt structures, is a result of local population growth. Development brings with it an increase in the amount of impervious surface in the environment. Any surface material (such as pavement) which water cannot penetrate is an impervious surface. Impervious surfaces collect runoff preventing it from filtering into the soil and surface sediments where the water is filtered removing some pollutants and stored. Runoff which comes from rain and snowmelt can collect pollutants in paved areas from manmade sources such as vehicles.
Paved areas also increase the risk of flash flooding. To lessen the potential hazard of flash flooding, stormwater management systems are incorporated into development plans. Such systems consist of a network of drains that collect runoff from streets and large parking areas and divert the runoff into local waterways. Sometimes the runoff is treated by a sewage treatment plant before discharge into a natural water resource. Unfortunately, during times of heavy runoff the additional water flow can overload a sewage treatment plant allowing a mixture of untreated sewage and stormwater to be dumped into the natural water resource.
Stormwater management systems work only as well as the weakest link in the system. Pipes can become clogged with debris, and if the initially installed pipes are too small, increased development may overwhelm the system. Thus, the design and maintenance of stormwater management systems is critical. It is always true that the greater the development in an area, the higher the risk of flash flooding.
PROBLEMS 
Hurricanes pose a great danger to the populations of the southern and eastern U.S. because such storms can, in just a few hours, deposit an amount of rainfall equivalent to several month's normal amount. The Potomac Mills Mall in Prince William County is one of the largest in the U.S. Its building and parking area represent 5,675411 ft^{2}. In 1972 hurricane Agnes dumped 14 inches of rain on the area over 3 days. The effects of this storm were described as the most costly disaster in the history of the Eastern Seaboard. The Potomac Mills Mall did not in 1972.
 Suppose Hurricane Sweetie hits Prince William County dumping 14 inches of
rain on the region in 72 hours. Assume that the Potomac MIlls
Mall was built on pasture and forested land which had a surface
geology that would hav absorbed most of the storm's runoff.
Calculate the maximum amount of extra runoff Potomac Mills Mall
will generate during Hurricane Sweetie. Express your answer in
cubic feet then convert it to cubic meters and liters.(1 ft^{3}
= 2.83 x 10 ^{2} m^{3}; 1 m^{3} = 1000
L).
 What is the resultant discharge from Potomac Mills Mall in liters per hour?
Compare your answer to the typical summer flow of the Potomac River
which is 100 million L/hr.
 Complete the table below by filling in the appropriate spaces to convert county area from acres to hectares and to convert impervious surface area from square feet to square meters. (1 acre = 0.4047 hectare)
Prince William County's Impervious Surface Area
Year  Housing Units  County area (acres)  County area (hectares)  Impervious surface area (ft^{2})  Impervious surface area (m^{2})  Area of county covered by impervious surface (%) (housing alone) 
1940  3545  222,080   6,696,000   0.07 
1950  5755  222,080   10,871,000   0.11 
1960  13''07  222,080   24,948,000   0.25 
1970  29,885  222,080   56,452,000   0.58 
1980  46,490  222,080   87,819,000   0.90 
1990  74,759  222,080   141,070,000   1.45 
1994  90,759  222,080   163,366,000   1.68 
 Using the table from question 3, plot the following graphs:
 housing units versus time
 impervious surface area (m^{2}) versus time
Interpret your graphs.
 Calculate average annual growth rates for both housing units and impervious surface area for the period 19401994. How do they compare?
 Using the growth rate for impervious surface area from 19401994, and starting
at 1994, calculate when 50% of Prince William County will be covered
with impervious surface from housing units. Speculate what life will
be like in Prince William County when 50% is covered with impervious
surface.

PROJECT 
Find U.S. Census data on growth for an Oregon county and make calculations similar to those you just did with Prince William County.
 What is the growth rate?
 What are its implications?
 Consider transportation. According to Statistical Abstract of the U.S.: 1996, there are approximately 0.6 motor vehicles per person in the U.S. Assuming the number of motor vehicles grows with the population, assess the impact of population growth on road congestion in this county.
 Is road construction and maintenance an unavoidable cost of pupulation growth?
 Should people who choose not to own cars pay for these new or expanded roads? Explain your answer by describing your reasoning.
 Examine the county's zoning ordinances (usually can be obtained from the city or town planner's office.) Determine if they are designed to promote or limit growth.

