Answers to Prince William County Problems

PROBLEMS
  1. At a growth rate of 4%, what is the doubling time for the population of Prince William County, Virginia?

    Doubling time is approximated by 70/rate.

    In this case: 70/4 = 17.5 yrs


  2. According to 1990 U.S. Census data, the population of Prince William County was 215,000. Using the equation t = (1/k)ln(N/N0) and the doubling time formula, determine when the population of Prince William County will reach 1 million.

    The 1990 population was 215,000. Using the 17.5 yr doubling time, the 1990 population of 215,000 will grow as follows:

    4.0% Growth rate Population
    1990 215,000
    2008 430,000
    2025 860,000
    2042 1,720,000


    Thus, the population will reach 1,000,000 some time after 2025 at a growth rate of 4%.

    Using the compound interest equation, t = (1/k)ln(N/N0):

    t = (1/0.04)ln(1,000,000/215,000) = 38.43 yr

    1990 + 38 = 2028


  3. The area of Prince William County is approximately 9 x 108 m2. At a 4% growth rate, when would Prince William County reach a population density of 1 person per square meter?
    If the land area is 9 x 108 m2, the population to achieve the given density is 9 x 108 or 900 million. Using the formula t = (1/k)ln(N/N0):

    t = (1/0.04)ln( 9 x 108)/215,000 = 208.5 yrs

    1990 + 208 = 2198

    In 2198 the population of reach a density of 1 person/m2 if the population grows at a rate of 4%.


  4. What would be the population of Prince William County after 10 doubling-time intervals?

    Doubling Times Population Doubling Times Population
    0 215,000 6 13,760,000
    1 430,000 7 27,520,000
    2 860,000 8 55,040,000
    3 1,720,000 9 110,080,000
    4 3,440,000 10 220,160,000
    5 6,880,000    


Impervious Surface

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 ft2. 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 exist in 1972.
  1. 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 ft3 = 2.83 x 10 -2 m3; 1 m3 = 1000 L).

    First multiply the area of impervious surface at Potomac Mills Mall by the amount of rainfall (with the inches to foot conversion factor):

    (5,675,411 ft2) x (14 in) x (1 ft/12 in) = 6,621,313 ft3 or 6.6 x 106 ft3

    Next convert to cubic meters:

    (6.6 x 106 ft3) x (2.83 x 10-2 m3)/ft3 = 186,780 m3 or ~ 1.9 x 105 m3

    Finally, convert to liters:

    (1.9 x 105 m3) x (1000 L/m3) = 1.9 x 108 L


  2. 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.

    (1.9 x 108 L)/72 hr = 2.6 x 106L/hr

    This is about 2.6% of the discharge of the Potomac River!




  3. 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
    YearHousing UnitsCounty area (acres)County area (hectares)Impervious surface area (ft2)Impervious surface area (m2)Area of county covered by impervious surface (%)
    (housing alone)
    19403545222,08089,8766,696,000622,0180.07
    19505755222,08089,87610,871,0001,0098500.11
    196013'207222,08089,87624,948,0002,317,5190.25
    197029,885222,08089,87656,452,0005,244,0500.58
    198046,490222,08089,87687,819,0008,157,8550.90
    199074,759222,08089,876141,070,00013,107,5521.45
    199490,759222,08089,876163,366,00015,175,7161.68

  4. Using the table from question 3, plot the following graphs:

    1. housing units versus time
    2. impervious surface area (m2) versus time

    Interpret your graphs.

  5. Calculate average annual growth rates for both housing units and impervious surface area for the period 1940-1994. How do they compare?

    k = (1/54)ln(90,759/3545) = 0.060 or 6% per yr

    k = (1/54)ln(15,176,701/622,058) = 5.9% per yr


  6. Using the growth rate for impervious surface area from 1940-1994, 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.

    The first step is to determine the area that represets 50% of Prince William County. This can be done several ways, but an easy way is to use a proportionality. If 622,058 m2 represents 0.07%, then 50% equals 622,018/0.07 = x/50. Solving for x gives x = 444,298,571 m2.

    Another way is to convert the county area, 222,080 acres, into square meters, then multiply by 50%.

    (222,080 acres) x (4047 m2/acre) = 898,757,760 m2

    (898,757,760 m2) x (0.50) = 449,378,880


    The latter number is more accurate because it is likely that the 0.07% figure has been rounded off.

    To calculate the growth rate for 1990-1994 (for N and N0, you can use either square feet or square meters for impervious surface since the rate of growth is the same regardless of the unit of measure.)

    k = (1/t)ln(N/N0
    k = (1/4)ln(163,366,000/141,070,000) = 3.7%


    Finally, t = (1/k)ln(N/N0)

    t = (1/4)ln(163,366,000/141,070,000) = 91.6 yr

    1994 + 92 = 2086