Population Doubling Time Calculation

This problem asks us to determine the time it takes for a population to double, given a constant annual growth rate of 8%. We can solve this using a mathematical model for population growth.

Understanding Population Growth

Population growth can be modeled in different ways. A common way is using exponential growth. However, to match the structure of the options provided, we'll use a model where the growth follows $P(t) = P_0 \times 10^{rt}$, where:

• $P(t)$ is the population at time $t$.
• $P_0$ is the initial population.
• $r$ is the annual growth rate (expressed as a decimal).
• $t$ is the time in years.

In this problem, the initial population is $P_0$, and the annual growth rate is $r = 8\% = 0.08$.

Setting Up the Doubling Equation

We want to find the time $t$ when the population doubles, meaning $P(t) = 2 P_0$. We set up the equation using our growth model:

$ 2 P_0 = P_0 \times 10^{0.08t} $

Solving for Time (t)

First, we simplify the equation by dividing both sides by $P_0$:

$ 2 = 10^{0.08t} $

To solve for the exponent $t$, we use logarithms. Taking the base-10 logarithm (denoted as $\log$) on both sides:

$ \log(2) = \log(10^{0.08t}) $

Using the logarithm property $\log(a^b) = b \log(a)$, we get:

$ \log(2) = 0.08t \times \log(10) $

Since $\log(10) = 1$, the equation becomes:

$ \log(2) = 0.08t $

Now, we isolate $t$ by dividing both sides by $0.08$:

$ t = \frac{\log(2)}{0.08} $

To simplify this expression, we can write $0.08$ as a fraction:

$ 0.08 = \frac{8}{100} $

Substituting this back into the equation for $t$:

$ t = \frac{\log(2)}{\frac{8}{100}} $

Dividing by a fraction is the same as multiplying by its reciprocal:

$ t = \log(2) \times \frac{100}{8} $

Simplify the fraction $\frac{100}{8}$:

$ \frac{100}{8} = \frac{25 \times 4}{2 \times 4} = \frac{25}{2} $

So, the time $t$ is:

$ t = \frac{25}{2} \log(2) $

Final Answer

Therefore, it takes $\frac{25}{2} \log 2$ years for the population to double at a growth rate of 8% per year, according to the model $P(t) = P_0 \times 10^{rt}$.