I can't seem to get away from these log problems!
At the end of last year, the population of Jason's hometown was approximately 75,000 people. The population is growing at the rate of 2.4% each year.
a) Write a function that models the growth of this city.
This answer is f(x) = 75,000(1.024)^x
b) How many years will it take for the population to reach 100,000 people?
Is the answer 12 or 13 years.......how can you tell because it's between the years!
y= 75000(1.024)* in 13 years.
How did you solve this -- I can do it two different ways and in one get 12.something and in another get 13!
You want
75,000(1.024)^x = 100,000
(1.024)^x = 4/3
x log1.024 = log(4/3)
x = log(4/3) / log1.024 = 0.1249/0.0103 = 12.13
To determine whether the answer is 12 or 13 years, we need to test each option and verify when the population reaches 100,000 people.
Let's start by evaluating the function for 12 years:
f(12) = 75,000(1.024)^12
≈ 75,000(1.3228)
≈ 99,210
The population at 12 years is approximately 99,210.
Now, let's evaluate the function for 13 years:
f(13) = 75,000(1.024)^13
≈ 75,000(1.34812)
≈ 101,109
The population at 13 years is approximately 101,109.
Since the population exceeds 100,000 after 12 years but is still below 100,000 after 13 years, we can conclude that it takes exactly 13 years for the population to reach 100,000 people.