A population doubles every 30 years. Assuming exponential growth find the following:

annual growth rate: %
continuous growth rate: %

2 = 1(1+i)^20

take 20th root of both sides
1.03526 = 1+i
i = .03526

the rate is 3.526%

continuous growth:
2 = e^(20k)
ln 2 = 20k(lne)
k = ln2/20 = .03466 or 3.466%

Well, if a population doubles every 30 years, that means it's growing pretty fast! Let's take a look at the annual growth rate first.

To find the annual growth rate, we can use the formula

Annual growth rate = (2^(1/30) - 1) * 100%

If we plug this into a calculator, we get an annual growth rate of approximately 2.26%. Wow, that's quite a growth spurt every year!

Now, let's move on to the continuous growth rate. Since we're dealing with exponential growth, we can use the formula

Continuous growth rate = ln(2)/30 * 100%

After some clown math, we find that the continuous growth rate is about 2.2909%. That's even more growth than the annual rate! Those populations sure know how to clown around when it comes to multiplying.

To find the annual growth rate, we can use the formula for exponential growth:

N(t) = N₀ * (1 + r)^t

Where:
N(t) = population at time t
N₀ = initial population
r = annual growth rate (as a decimal)
t = time in years

Since the population doubles every 30 years, we can say:

N(30) = 2 * N₀

Substituting the values into the formula:

2 * N₀ = N₀ * (1 + r)^30

Simplifying the equation:

2 = (1 + r)^30

Taking the 30th root of both sides:

2^(1/30) = 1 + r

Subtracting 1 from both sides:

2^(1/30) - 1 = r

Using a calculator, the result is approximately 0.023. Therefore, the annual growth rate is 0.023 or 2.3%.

To find the continuous growth rate, we can use the formula:

r_continuous = ln(1 + r)

Where:
r_continuous = continuous growth rate
r = annual growth rate (as a decimal)

Substituting the value into the formula:

r_continuous = ln(1 + 0.023)

Using a calculator, the result is approximately 0.0228. Therefore, the continuous growth rate is 0.0228 or 2.28%.

To find the annual growth rate, you can use the formula:

Annual Growth Rate = (1 + r)^n - 1

Where:
- r is the annual growth rate (in decimal form)
- n is the number of periods (in this case, years)

In this scenario, the population doubles every 30 years, meaning that the population grows by a factor of 2.

To find the annual growth rate, let's plug in the values into the formula:

2 = (1 + r)^30

Now, solve for r by taking the 30th root of both sides:

(1 + r) = 2^(1/30)

Now subtract 1 from both sides:

r = 2^(1/30) - 1

Multiply the result by 100 to convert it to a percentage:

r = (2^(1/30) - 1) * 100

This will give you the annual growth rate.

To find the continuous growth rate, you can use the formula:

Continuous Growth Rate = ln(2) / t

Where:
- ln represents the natural logarithm
- 2 represents the growth factor (in this case, the population doubles, so the factor is 2)
- t represents the time period (in this case, 30 years)

Plugging in the values:

Continuous Growth Rate = ln(2) / 30

Evaluate this expression to get the continuous growth rate.

So, to summarize:
- To find the annual growth rate, calculate (2^(1/30) - 1) * 100.
- To find the continuous growth rate, calculate ln(2) / 30.