Questions: The population of a city has been increasing by 2% annually. The sign shown is from the year 2000 CITY LIMIT BROOKFIELD pop. 315,000 What will the population be in 2020? Round your answer to the nearest thousand. The population will be about

Solution Steps

To solve this problem, we need to calculate the population of the city in 2020 given that it has been increasing by 2% annually since the year 2000. We can use the formula for exponential growth:

\[ P(t) = P_0 \times (1 + r)^t \]

where:

• \( P(t) \) is the population at time \( t \)
• \( P_0 \) is the initial population (315,000 in the year 2000)
• \( r \) is the annual growth rate (2% or 0.02)
• \( t \) is the number of years since the initial time (2020 - 2000 = 20 years)

We will plug these values into the formula to find the population in 2020.

Let \( P_0 = 315,000 \) be the initial population in the year 2000. The annual growth rate is \( r = 0.02 \) (or \( 2\% \)). The time period from 2000 to 2020 is \( t = 20 \) years.

We use the formula for exponential growth:

\[ P(t) = P_0 \times (1 + r)^t \]

Substituting the known values:

\[ P(20) = 315,000 \times (1 + 0.02)^{20} \]

Calculating \( (1 + 0.02)^{20} \):

\[ (1.02)^{20} \approx 1.485947 \]

Now, substituting this back into the equation:

\[ P(20) \approx 315,000 \times 1.485947 \approx 468,073.4297 \]

Rounding \( 468,073.4297 \) to the nearest thousand gives:

\[ \text{Population in 2020} \approx 468,000 \]

Final Answer