The population of a city was 60,000 in the year 2005 and increased by 2.5% every year.
a) Write a formula for the population size depending on time
b) What was the population in 2015?
c) In what year will the population double?
a) p = 60000 * (1 + .025)^t ... t in years
b) plug in 10 for t ... solve for p
c) plug in 120000 for p ... solve for t
a) To find a formula for the population size depending on time, we can use the concept of exponential growth. The formula for exponential growth is:
Population = Initial Population * (1 + Growth Rate)^Time
In this case, the initial population is 60,000, and the growth rate is 2.5% or 0.025 (expressed as a decimal). The time would represent the number of years since the initial population.
So, the formula for the population size depending on time would be:
Population = 60,000 * (1 + 0.025)^Time
b) To find the population in the year 2015, we need to calculate the number of years since 2005. Since 2015 is 10 years after 2005, we can substitute Time = 10 into the formula:
Population = 60,000 * (1 + 0.025)^10
Population = 60,000 * (1.025)^10
Calculating the population using a calculator or spreadsheet, we find that the population in 2015 was approximately 79,486.
c) To determine the year in which the population will double, we need to find the value of Time when the Population is twice the Initial Population (60,000 * 2 = 120,000).
Setting Population = 120,000 in the formula, we get:
120,000 = 60,000 * (1 + 0.025)^Time
Dividing both sides of the equation by 60,000 and rearranging the equation, we have:
2 = (1.025)^Time
To solve for Time, we need to take the logarithm of both sides. Assuming a logarithm with base 10, the equation becomes:
log10(2) = log10((1.025)^Time)
Using the logarithmic property of exponents (log(ab) = b * log(a)), the equation further simplifies to:
log10(2) = Time * log10(1.025)
Finally, we can solve for Time by dividing both sides by log10(1.025):
Time = log10(2) / log10(1.025)
Using a calculator, we find that Time is approximately 28.2 years.
Therefore, the population will double in approximately 28.2 years from the initial year of 2005, which corresponds to the year 2005 + 28 = 2033.