Modeling Population Growth in Austin, Texas

MODELING WITH MATHEMATICS

Population Growth in Austin, Texas

  • The population of Austin, Texas, can be modeled using the following function:
    • y = 494.29 (1.03)^{t}
    • Where:
    • y represents the population in thousands,
    • t is the number of years since the beginning of the decade.

Analysis of the Function

A. Nature of the Function

  • Exponential Growth vs. Exponential Decay
    • The function represents exponential growth because:
    • The base of the exponent (1.03) is greater than 1.

B. Annual Percent Change

  • Annual Percent Increase
    • The annual percent increase in the population can be calculated using the formula for exponential growth:
    • If the function is of the form y = a(1 + r)^{t} ,
    • Then the annual percent increase r is given by:
      • r = 1.03 - 1 = 0.03
    • Therefore, the annual percent increase is:
      • 0.03 \times 100 = 3\%

C. Population Milestone

  • Population of 590,000

    • To find out when the population reaches approximately 590,000:
    • Set the function equal to 590,000:
      • 590 = 494.29(1.03)^{t}
    • Solving for t :
      • 590 / 494.29 = (1.03)^{t}
      • This leads to:
      • (1.03)^{t} \approx 1.194
    • Take the logarithm of both sides:
      • t \cdot \log(1.03) = \log(1.194)
      • Therefore:
      • t \approx \frac{\log(1.194)}{\log(1.03)}
    • Calculating the above will yield the approximate value of t , which indicates how many years after the start of the decade the population was about 590,000.

  • Approximate Year

    • Using the above calculation, the population is about 590,000 approximately x years after the start of the decade (where x is the calculated value).