If the rate of change of the population is proportional to the population and the population of a country doubled in 35 years, what is its yearly rate of increase?

I don't remember how to do this at all, so I really don't know how to start the problem.

It also says assume a constant rate of increase.

we are told that p doubles every 35 years. That means

p = Po * 2^(t/35)

Each year it increases by a factor of 2^(1/35) = 1.02, or 2% per year.

To solve this problem, we can use the concept of exponential growth, where the rate of change is proportional to the population. The general formula for exponential growth is given by:

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

Here, P(t) represents the population at time t, P0 is the initial population, k is the growth constant, and e is the base of the natural logarithm, approximately equal to 2.71828.

Given that the population of the country doubled in 35 years, we can write this as:

2P0 = P0 * e^(k*35)

Simplifying this equation, we find:

2 = e^(35k)

Now, to find the yearly rate of increase, we need to solve for k.

First, take the natural logarithm of both sides of the equation:

ln(2) = 35k * ln(e)

Since ln(e) equals 1, we can simplify further:

ln(2) = 35k

Now, divide by 35 to isolate k:

k = ln(2) / 35

Finally, to find the yearly rate of increase, multiply k by 100 to convert it to a percentage:

Rate of increase = k * 100 = (ln(2) / 35) * 100

Using a calculator, we can evaluate this expression to get the final answer.