Q.In 2005, for each of the 14 million people present in a country, 0.028 were born and 0.008 died during the year. Using exponential equation, the number of people present in 2015 is predicted as:
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Start your 14-day free trial to unlock the full solution →The population in 2015 is predicted to be 17 million, calculated using the exponential growth equation based on the intrinsic rate of natural increase derived from the given birth and death rates.
In population ecology, when resources are unlimited and there are no predators or environmental resistance, a population tends to grow exponentially. This means that the rate of population increase itself accelerates over time, leading to a J-shaped growth curve. This model is often observed in newly colonised habitats or during the initial phases of population growth before resource limitations become significant.
The core idea behind exponential growth is that the population's growth rate is proportional to its current size. The larger the population, the more individuals are available to reproduce, leading to an even faster increase in numbers. This continuous growth is mathematically represented using the base of the natural logarithm, $e$.
To quantify this growth, we first need to determine the intrinsic rate of natural increase, denoted by $r$. This value represents the per capita growth rate of the population under ideal conditions. It is calculated by subtracting the per capita death rate from the per capita birth rate.
- Per capita birth rate (b) = $0.028$
- Per capita death rate (d) = $0.008$
Therefore, the intrinsic rate of natural increase ($r$) is:
$$r = b - d$$
$$r = 0.028 - 0.008$$
$$r = 0.020 \text{ per year}$$
This means that, on average, for every individual in the population, there is a net increase of $0.020$ individuals per year.
The exponential growth equation is given by:
$$N_t = N_0 e^{rt}$$
Where:
- $N_t$ is the population size at time $t$.
- $N_0$ is the initial population size (at time $t=0$).
- $e$ is the base of the natural logarithm, approximately $2.71828$.
- $r$ is the intrinsic rate of natural increase.
- $t$ is the time period over which the growth occurs.
Now, let's identify the values we have for our calculation: …
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