The table gives the population of a particular country, in millions, for the years 1900 - 2000. ...

Question:

The table gives the population of a particular country, in millions, for the years 1900 - 2000.

Year Population
1900 76
1910 92
1920 106
1930 123
1940 131
1950 150
1960 179
1970 203
1980 227
1990 250
2000 275

Use this model to predict the population in the year 2010.

Use this model to predict the population in the year 2020.

Population Models:

There are various kinds of population models that can be used to predict the future value of a population. No model will be able to an accurate picture when we use it to predict values further and further in the future. This is because no matter how accurate a model is, it can never account for all the future events that could affect the population,.

Answer and Explanation:

We can use an exponential model to predict the population.

$$P(t)=P(o)e^{kt} $$


Here, P(t) is the population after t years when the population at t=0 was P(o). k is the rate of growth.

Now, using the above model we want to predict the population in 2010 and 2020.

To predict the population, we first need to complete our model. We need to decide the value of P(o) and the value after t years, P(t), using the information given to find the value of k.

Even though we can pick any two values from the data given, we pick up the values in 1990 and 2000 for accuracy. So, P(o) is 250 and the population after t=10 years is P(10)=275. The value of k is found as follows.

$$\begin{align} &P(10)=250e^{10k}=275\\ &10k=\ln \left ( \frac{275}{250} \right )\\ \therefore &k\approx0.009531 \end{align} $$


Our model now is:

$$P(t)=250e^{0.009531t} $$


The population figures in 2010 (t=20) and 2020 (t=30) will now be:

$$\begin{align} P(20)&=250e^{0.009531*20}\\&\approx 302.5\,\text{million}\\ P(30)&=250e^{0.009531*30}\\&\approx 332.75\,\text{million} \end{align} $$


Learn more about this topic:

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Exponential Growth: Definition & Examples

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