A city started with a population of 16,000, and it grew at the rate of 7% per year. The current population is 27,490. How many years did it take to reach this size

Question

growth and decay for algebra 2

Answer 1

Answer 2

Hi there,

Try using the following equation for continuous exponential growth:

P(t) = P₀ e^kt

where P₀ is the initial amount, k is rate (in decimal form!), t is the amount of time that has passed (in this case, in years), and P(t) is the amount after t years.

From the prompt, we know

1. the initial population is 16,000, so P₀=16,000
2. the rate is 7%, so k=0.07
3. the current population or "new" population is 27,490, so P(t)=27,490
4. we don't know how much time has gone by, so we need to solve for t

Now all you have to do is plug in the values in the appropriate places. So, we get

27,490 = 16,000e^0.07t

Dividing both sides by 16,000, resulting in

(27,490/16,000) = e^0.07t

To get t on its own, we'll have to get rid of that e. Take the natural log of both sides of the equation. We get

ln(27,490/16,000) = ln(e^0.07t)

ln(27,490/16,000) = 0.07t

Divide both sides by 0.07. Plugging this into your calculator, you should get

t = 7.73 years

Hope this helps!

Answer 3

Using the formula for exponential growth and decay we can solve this problem. The formula is P = P₀e^rt where P₀ is the initial value (16000), P is the final value (27490), r is the rate (7% or 0.07 growth per year), and t is the time in years. Plugging these numbers in we have the equation 27490 = 16000*e^(0.07)t. We now need to solve for t, shown in the following steps:

1. 27490 = 16000*e^(0.07)t
2. 27490/16000 = e^(0.07)t
3. ln(1.718125) = ln(e^(0.07)t)
4. 0.541233 = 0.07t
5. t = 0.541233/0.07 = 7.73 years or about 7 years 9 months.

Answer 4

Hello, Efef,

At the start (Year 0), the population is 16,000. The general formula for finding the population at any year, x, afterwards is:

(16,000)*(1.07)^x

You can graph this equation or make a table, as I've done here:

It looks like 8 years is the correct number for the population to reach 2790. Off by 1, but I'll blame the census.

Bob