The population of a city is growing at an average rate of 4.5% per year. In 1994, the population was 30000.
A) write an equation that models the growth of the city.
B) use your equation to determine the population of the city in 2015.
C) Determine the year during which the population will have doubled
A) To write an equation that models the growth of the city, we can use the formula:
Population = Initial Population * (1 + Growth Rate)^(Number of Years)
Let P represent the population, P0 the initial population, r the growth rate, and t the number of years. In this case, P0 is given as 30000 and the growth rate is 4.5% or 0.045. The equation becomes:
P = 30000 * (1 + 0.045)^t
B) To determine the population in 2015, we need to find the value of t when t = 2015 - 1994 = 21. Plugging this into the equation:
P = 30000 * (1 + 0.045)^21
P ≈ 30000 * 1.9857
P ≈ 59571
Therefore, the population of the city in 2015 is approximately 59571.
C) To determine the year during which the population will have doubled, we need to solve the equation:
2P0 = P0 * (1 + 0.045)^t
Dividing both sides by P0:
2 = (1 + 0.045)^t
Taking the logarithm of both sides:
log(2) = t * log(1 + 0.045)
Using logarithm properties:
t = log(2) / log(1.045)
t ≈ 15.85355
Therefore, the population will have doubled in approximately 16 years, which would be in the year 1994 + 16 = 2010.