A population doubles every 18 years. Assuming exponential growth find the following:

(a) The annual growth rate:

(b) The continuous growth rate is

(a) The annual growth rate can be found using the formula for exponential growth:

Growth Rate = (ln(2))/(Doubling Time)

Given that the population doubles every 18 years, the doubling time is 18 years. Therefore, the annual growth rate is:

Growth Rate = (ln(2))/(18) ≈ 0.0387

(b) The continuous growth rate can be found using the formula:

Continuous Growth Rate = ln(2)/Doubling Time

Using the given doubling time of 18 years, the continuous growth rate is:

Continuous Growth Rate = ln(2)/18 ≈ 0.0387

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

P(t) = P0 * e^(rt)

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

In this case, we know that the population doubles every 18 years. This means that after 18 years, the population is twice as large as the initial population:

P(18) = 2 * P0

Substituting these values into the exponential growth formula:

2 * P0 = P0 * e^(18r)

Dividing both sides of the equation by P0:

2 = e^(18r)

To solve for r, we need to take the natural logarithm of both sides:

ln(2) = ln(e^(18r))

Using the logarithmic property, we can move the exponent down:

ln(2) = 18r * ln(e)

Since ln(e) equals 1, the equation simplifies to:

ln(2) = 18r

Now, we can solve for r by dividing both sides by 18:

r = ln(2) / 18

Using a calculator, we can find the value of r:

r ≈ 0.03873

Therefore, the annual growth rate is approximately 0.03873 or 3.873% per year.

(b) To find the continuous growth rate, we can use the relationship between the annual growth rate (r) and the continuous growth rate (k):

k = ln(1 + r)

Substituting the value we found for r:

k = ln(1 + 0.03873)

Using a calculator, we can find the value of k:

k ≈ 0.03811

Therefore, the continuous growth rate is approximately 0.03811 per year.

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

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, and t is the time in years.

(a) The annual growth rate (r) can be obtained by rearranging the formula:

P(t) = P₀ * e^(rt)
P(t)/P₀ = e^(rt)
ln(P(t)/P₀) = ln(e^(rt))
ln(P(t)) - ln(P₀) = rt
r = (ln(P(t)) - ln(P₀))/t

In this case, we are given that the population doubles every 18 years. So, P(t) = 2P₀, where P₀ is the initial population. Plugging this into the formula, we have:

r = (ln(2P₀) - ln(P₀))/18

Simplifying further:

r = ln(2)/18 ≈ 0.038
Therefore, the annual growth rate is approximately 0.038 or 3.8%.

(b) The continuous growth rate can be found using the relationship between continuous growth rate (k) and annual growth rate (r):

k = ln(1 + r)

Plugging in the value of r calculated in part (a), we have:

k = ln(1 + 0.038) ≈ 0.037 or 3.7%

The continuous growth rate is approximately 0.037 or 3.7%.

annual: 2^(1/18) = 1.03926 or 3.93%

continuous: 1/18 ln2 = 0.0385 or 3.85%