How long will it take for the population of a certain country to double if its annual growth rate is 1.5%? Round to the nearest year. Use the exponential growth model P(t) = P0e^kt

Two ways to do it:

each year multiply by 1.015 (like compound interest)
1.015^n = 2
n log 1.015 = log 2
n = log 2 / log1.015 = 46.55 years

or do it the way they said to do it with continuous growth
dy/dt = .015y
y = yi e^.015 t
y/yi = 2 = e^.015 t
ln 2 = .015 t
t = 46.2

To determine how long it will take for the population of a certain country to double, we can use the exponential growth model. Let's break down the steps to find the answer:

Step 1: Understand the formula
The formula we will use is the exponential growth model, given by:
P(t) = P0e^(kt)

Where:
P(t) is the population at time t
P0 is the initial population
e is the mathematical constant approximately equal to 2.71828
k is the growth rate
t is the time elapsed

Step 2: Set up the equation
In this case, we want to find the time it takes for the population to double. So, we need to find the time t when P(t) = 2P0 (twice the initial population).

P(t) = P0e^(kt)
2P0 = P0e^(kt)

Step 3: Simplify the equation
To solve for t, we need to eliminate P0 on both sides of the equation.

2 = e^(kt)

Step 4: Take the natural logarithm of both sides
To get rid of the exponential term e^(kt), we can take the natural logarithm (ln) of both sides.

ln(2) = ln(e^(kt))

Step 5: Simplify the equation further
Using the logarithm property ln(e^x) = x, we can simplify the equation.

ln(2) = kt

Step 6: Solve for t
Now, we can solve for t by dividing both sides of the equation by k.

t = ln(2) / k

Step 7: Plug in values and calculate
In this case, the annual growth rate is 1.5%, which can be written as 0.015 in decimal form. So, we substitute k = 0.015 into the equation.

t = ln(2) / 0.015

Using a calculator, we find that ln(2) is approximately 0.6931.

t ≈ 0.6931 / 0.015 ≈ 46.21

Therefore, it will take approximately 46 years for the population of the country to double based on a 1.5% annual growth rate. Rounded to the nearest year, it will take 46 years.

To determine how long it will take for the population of a certain country to double, we can use the exponential growth model:

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

Where:
P(t) is the population at time t,
P0 is the initial population,
e is the base of the natural logarithm (approximately 2.71828),
k is the continuous growth rate, and
t is the time (in years).

In this case, the annual growth rate is given as 1.5%. We need to convert this percentage into a decimal by dividing it by 100:

k = 1.5% / 100
k = 0.015

Let's denote the initial population as P0 and the time it takes to double the population as t2.

Given that the initial population is P0, and the final population will be 2 * P0 (since we want to find when it doubles), we can write the equation as:

2 * P0 = P0 * e^(0.015t2)

Simplifying the equation:

2 = e^(0.015t2)

Taking the natural logarithm on both sides:

ln(2) = ln(e^(0.015t2))

Using the rule of logarithms, we can simplify further:

ln(2) = 0.015t2 * ln(e)

Since ln(e) equals 1, we have:

ln(2) = 0.015t2

Now, we can solve for t2:

t2 = ln(2) / 0.015

Using a calculator, we find:

t2 ≈ 46.2 years

Therefore, it will take approximately 46 years for the population of the country to double with an annual growth rate of 1.5%.