a town has a population of 8400 in 1990. Fifteen years later, it's population had grown to 12 500. Assuming that the population continues to grow at the same exponential rate, when will the population reach 20 000?

Please help im trying to answer this question for past 2 hours

it grew by a factor of 125/84 = 1.488 in 15 years. That is an annual rate of 1.488^(1/15) = 1.02685

So, if t is the number of years after 1990, the population can be found using

P(t) = 8400 * 1.02685^t

To find when P(t) will be 20000, just solve for t in

8400 * 1.02685^t = 20000
1.02685^t = 2.38095
t = log(2.38095)/log(1.02685) = 32.74

So, in the 33rd year after 1990 -- 2023 -- the population will reach 20000

Thanks a lot.

yeeet

To find the year when the population of the town reaches 20,000, we can use the formula for exponential growth.

The formula for exponential growth is: P(t) = P₀ * e^(rt)

Where:
P(t) is the population at time t
P₀ is the initial population
e is Euler's number (approximately 2.71828)
r is the growth rate
t is the time in years

We know that the population in 1990 (t = 0) was 8,400 (P₀), and after 15 years (t = 15) it had grown to 12,500 (P(t) = 12,500).

We can use this information to find the growth rate (r).

12,500 = 8,400 * e^(15r)

To solve for r, divide both sides of the equation by 8,400:

12,500 / 8,400 = e^(15r)

1.4881 = e^(15r)

Next, take the natural logarithm of both sides to isolate the exponent:

ln(1.4881) = 15r ln(e)

Using a calculator, ln(1.4881) ≈ 0.3988 and ln(e) = 1. Therefore:

0.3988 = 15r

Now solve for r by dividing both sides by 15:

r ≈ 0.0266

Now that we have the growth rate, we can find the time it takes for the population to reach 20,000.

20,000 = 8,400 * e^(0.0266t)

To solve for t, divide both sides of the equation by 8,400:

20,000 / 8,400 = e^(0.0266t)

2.3809 = e^(0.0266t)

Take the natural logarithm of both sides to isolate the exponent:

ln(2.3809) = 0.0266t ln(e)

Using a calculator, ln(2.3809) ≈ 0.8687 and ln(e) = 1. Therefore:

0.8687 = 0.0266t

To solve for t, divide both sides by 0.0266:

t ≈ 32.65

Since t represents the number of years after 1990, to find the year when the population reaches 20,000, add 32.65 to 1990:

1990 + 32.65 ≈ 2022.65

So, the population of the town is projected to reach 20,000 in the year 2023.