Hi,
Can you please help me. Thank-you.
The present population of a town is 42000.
a) Determine the population 4 years ago if the growth rate has been a constant 4%/year.
b) Determine the number of complete years that have passed since the town first had a population of less than 13000.
a) 42,000 = P(1.04)^4 making the population 4 years ago 35,902.
b) 42,000 = 13,000(1.04)^n making n = 29.9 or 29 whole years.
Of course! I'd be happy to help you with your questions.
a) To determine the population 4 years ago with a constant growth rate of 4% per year, you can use the formula for compound interest:
F = P(1 + r)^n
Where:
F is the final population after n years,
P is the initial population,
r is the growth rate, and
n is the number of years.
Given that the present population is 42000 and the growth rate is 4%, we can substitute the values into the formula and solve for P:
F = 42000
r = 4% = 0.04
n = 4
42000 = P(1 + 0.04)^4
Dividing both sides of the equation by (1 + 0.04)^4:
P = 42000 / (1.04^4)
Using a calculator, we find that P is approximately 36005.19.
Therefore, the population 4 years ago was approximately 36005.
b) To determine the number of complete years that have passed since the town first had a population of less than 13000, you can use the same formula:
F = P(1 + r)^n
Given that the initial population is less than 13000, we can start with a hypothetical population of 13000 and solve for n:
F = 13000
P = 13000
r = 4% = 0.04
13000 = 13000(1 + 0.04)^n
Dividing both sides of the equation by 13000 and simplifying:
1 = (1.04)^n
Taking the logarithm of both sides of the equation:
log(1) = log(1.04)^n
0 = n*log(1.04)
n = 0 / log(1.04)
Using a calculator, we find that n is approximately 0.
This means that no complete year has passed since the town first had a population of less than 13000.
I hope this helps! Let me know if you have any other questions.
Of course, I'd be happy to help you!
a) To determine the population 4 years ago if the growth rate has been a constant 4% per year, you can use the formula for exponential growth:
Population (t) = Population (0) * (1 + Growth Rate)^t
Where:
- Population (t) is the population at time t
- Population (0) is the initial population
- Growth Rate is the rate at which the population is growing
- t is the time in years
In this case, the initial population (Population (0)) is 42000, the growth rate is 4% (or 0.04), and you want to find the population 4 years ago. So, we can plug these values into the formula:
Population (t) = 42000 * (1 + 0.04)^t
Now, since we want to find the population 4 years ago, we need to substitute t with -4. This is because a negative value for t represents the number of years in the past:
Population (-4) = 42000 * (1 + 0.04)^-4
Using a calculator, you can evaluate this expression to find the population 4 years ago.
b) To determine the number of complete years that have passed since the town first had a population of less than 13000, you need to find the year when the population first dropped below 13000 and then subtract it from the current year.
Given that the present population is 42000, you need to determine the year when the population was less than 13000. We can set up an equation using the formula for exponential growth:
13000 = Population (0) * (1 + Growth Rate)^t
Now, rearrange the equation to solve for t:
t = log[1 + Growth Rate](13000 / Population (0))
Plug in the values: Population (0) = 42000, Growth Rate = 0.04, and 13000 for the population when t years have passed. You can use a logarithm function on a calculator that supports base 1 + Growth Rate to find the value of t.
Once you have the value of t, you can subtract it from the current year to determine the number of complete years that have passed since the town first had a population of less than 13000.
I hope this helps! Let me know if you have any further questions.