If the population of the small town is currently 13,000 people, and it's decreasing at 4% each year, how many years will it take for the population to reach 8,500 people?
13000(.96)^n = 8500
.96^n = 8500/13000 = 17/26
take log of both sides, and use log rules
n log .96 = log17 - log26
n = (log17 - log26)/log.96 = appr .......
Well, let's calculate that! If the population is decreasing by 4% each year, that means it's losing 4% of its population annually. In other words, the population will go down to 96% of what it was the previous year.
So, to find out when the population will reach 8,500 people, we can set up an equation:
13,000 * 0.96^x = 8,500
Now, let me do some quick calculations here... (imaginary calculator noises)...
Ah, it looks like it will take approximately 9.83 years for the population to reach 8,500 people.
But before you start counting your chickens, remember that population statistics can be a bit unpredictable. It's like trying to predict how many clowns will fit into a tiny car. You can estimate, but you never quite know for sure!
To find out how many years it will take for the population to reach 8,500 people, we need to calculate the number of years it takes for the population to decrease from 13,000 to 8,500.
Step 1: Find the annual decrease in population.
Since the population is decreasing at a rate of 4% per year, we can find the annual decrease by multiplying 13,000 by 4% (or 0.04).
Annual decrease = 13,000 x 0.04 = 520 people per year.
Step 2: Calculate the number of years it takes to reach 8,500 people.
We need to divide the difference between the current population (13,000) and the target population (8,500) by the annual decrease.
Difference in population = 13,000 - 8,500 = 4,500 people.
Number of years = Difference in population / Annual decrease = 4,500 / 520 ≈ 8.65 years.
Therefore, it will take approximately 8.65 years for the population to reach 8,500 people.
To find out how many years it will take for the population to reach 8,500 people, we can use the formula for exponential decay:
P = P₀ * (1 - r)^t
Where:
P is the final population (8,500 people),
P₀ is the initial population (13,000 people),
r is the rate of decrease (4% or 0.04), expressed as a decimal,
t is the number of years.
We can rearrange the formula to solve for t:
t = log(P / P₀) / log(1 - r)
Now let's plug in the values and calculate:
t = log(8,500 / 13,000) / log(1 - 0.04)
Using a calculator, the value of t is approximately 10.65.
Since we can't have fractional years, we will round up, which means it will take approximately 11 years for the population to reach 8,500 people.