The population of a town is modeled by the equation P = 3485e^0.12t , where p represents the population as of the year 2000.
Part A: According to the model, what will the population of the town be in 2015?
Part B: In approximately what year will the population reach 57,000 people?
Part A:
To find the population in the year 2015, we need to substitute t = 15 (since 2015 is 15 years after 2000) into the equation:
P = 3485e^(0.12*15)
P = 3485e^1.8
P ≈ 3485 * 6.0496
P ≈ 21,074.432
Therefore, the population of the town in 2015 will be approximately 21,074 people.
Part B:
To find the year in which the population will reach 57,000 people, we need to set P = 57000 and solve for t:
57000 = 3485e^(0.12t)
57000/3485 = e^(0.12t)
16.362 = e^(0.12t)
ln(16.362) = ln(e^(0.12t))
ln(16.362) = 0.12t
t = ln(16.362) / 0.12
t ≈ 7.66
Therefore, the population of the town will reach 57,000 people in approximately the year 2007 + 7.66 = 2015.66, which is around the middle of 2015.