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 of the town in 2015, we need to substitute t = 2015 - 2000 = 15 into the equation:
P = 3485e^0.12t
P = 3485e^0.12(15)
P ≈ 3485e^1.8
Using a calculator, we find that P ≈ 18128.35
Therefore, the population of the town in 2015 is approximately 18,128.
Part B: To find approximately what year the population will reach 57,000 people, we need to solve the equation for t:
57,000 = 3485e^0.12t
Dividing both sides of the equation by 3485, we get:
e^0.12t ≈ 57,000/3485
Taking the natural logarithm of both sides, we have:
0.12t ≈ ln(57,000/3485)
t ≈ ln(57,000/3485) / 0.12
Using a calculator, we find that t ≈ 5.69
Adding t = 5.69 to the given year 2000, we have:
Year ≈ 2000 + 5.69 ≈ 2005.69
Therefore, the population will reach 57,000 people in approximately the year 2005.69, which can be rounded to 2006.