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Researchers are studying the population of a city. Based on current trends, they use this function to model the city's population x years from now.

y=3,700x+135,000

\tell me the cities current population, how much it is expected to increase or decrease each year, and how many years it will take for the number 172,000 to be reached.

1 answer

To analyze the population function given by

\[ y = 3,700x + 135,000, \]

we can interpret the components of this equation.

  1. Current population: When \( x = 0 \) (which represents the present year), the current population \( y \) can be calculated as follows:

    \[ y = 3,700(0) + 135,000 = 135,000. \]

    So, the current population of the city is 135,000.

  2. Annual increase: The coefficient of \( x \) in the equation (which is 3,700) represents the annual increase in the population. Therefore, the population is expected to increase by 3,700 each year.

  3. Years to reach a population of 172,000: To find out how many years it will take for the population to reach 172,000, we set \( y \) to 172,000 and solve for \( x \):

    \[ 172,000 = 3,700x + 135,000. \]

    Subtracting 135,000 from both sides gives:

    \[ 172,000 - 135,000 = 3,700x, \] \[ 37,000 = 3,700x. \]

    Now, we divide both sides by 3,700:

    \[ x = \frac{37,000}{3,700} = 10. \]

    Thus, it will take 10 years for the population to reach 172,000.

Summary:

  • Current population: 135,000
  • Annual increase: 3,700
  • Years to reach 172,000: 10 years