Calculating Population Doubling Time

To find the time it takes for the population to double, we can use the formula for exponential growth:

$ P(t) = P_0 (1 + r)^t $

Where:

• $P(t)$ is the population after $t$ years.
• $P_0$ is the initial population (5 million).
• $r$ is the annual growth rate (20% or 0.20).
• $t$ is the time in years.

We want to find the time $t$ when the population doubles, meaning $P(t) = 2 \times P_0$.

Population Doubling Equation

Setting up the equation for doubling:

$ 2 P_0 = P_0 (1 + 0.20)^t $

Divide both sides by $P_0$:

$ 2 = (1.20)^t $

Solving for Time (t)

To solve for $t$, we use logarithms:

1. Take the logarithm of both sides (natural log or base-10 log):

   $ \log(2) = \log((1.20)^t) $

2. Use the logarithm power rule ($\log(a^b) = b \log(a)$):

   $ \log(2) = t \times \log(1.20) $

3. Isolate $t$:

   $ t = \frac{\log(2)}{\log(1.20)} $

4. Calculate the values:

   $ t \approx \frac{0.3010}{0.0792} $

   $ t \approx 3.799 \text{ years} $

The calculated time is approximately 3.799 years. This value falls within the 3-4 years range.