The population of a town is modeled by the equation P=3485e0.12t%0D%0A%0D%0A=%0D%0A3485%0D%0A%0D%0A0.12%0D%0A%0D%0A, where P%0D%0A%0D%0A represents the population as of the year 2000.%0D%0A%0D%0APart A: According to the model, what will the population of the town be in 2015?%0D%0A%0D%0APart B: In approximately what year will the population reach 57,000 people?%0D%0A%0D%0A(1 point)%0D%0AResponses%0D%0A%0D%0A %0D%0A%0D%0APart A: 11,571 people in 2015%0D%0A%0D%0APart B: Around the year 2022%0D%0A%0D%0A %0D%0A%0D%0A Part A: 11,571 people in 2015 Part B: Around the year 2022 %0D%0A%0D%0A %0D%0A%0D%0APart A: 21,083 people in 2015%0D%0A%0D%0APart B: Around the year 2025%0D%0A%0D%0A %0D%0A%0D%0A Part A: 21,083 people in 2015 Part B: Around the year 2025 %0D%0A%0D%0A %0D%0A%0D%0APart A: 21,083 people in 2015%0D%0A%0D%0APart B: Around the year 2023%0D%0A%0D%0A %0D%0A%0D%0A Part A: 21,083 people in 2015 Part B: Around the year 2023 %0D%0A%0D%0A %0D%0A%0D%0APart A: 11,571 people in 2015%0D%0A%0D%0APart B: Around the year 2020
show your work
To find the population of the town in 2015, we can substitute t = 2015-2000 = 15 into the equation:
P = 3485e^(0.12*15)
Using a calculator, we can calculate that P is approximately 11,571 people.
To approximate the year when the population reaches 57,000 people, we need to solve the equation:
57,000 = 3485e^(0.12t)
Divide both sides of the equation by 3485:
e^(0.12t) = 57,000/3485
Take the natural logarithm of both sides:
ln(e^(0.12t)) = ln(57,000/3485)
0.12t = ln(57,000/3485)
Divide both sides by 0.12:
t = ln(57,000/3485) / 0.12
Using a calculator, we can approximate that t is approximately 22.89.
Since t represents the time since the year 2000, we add this to 2000 to find the approximate year when the population reaches 57,000 people.
Approximately, the year would be 2000 + 22.89 = 2022.89, which can be rounded to the year 2023.