Problem 14 - Algebra and Trigonometry

Question

In \(2000,\) the population of Greece was \(10,600,000,\) with projections of a population decrease of \(28,000\) people per year. In the same year, the population of Belgium was \(10,200,000,\) with projections of a population decrease of \(12,000\) people per year. (Source: United Nations) According to these projections, when will the two countries have the same population? What will be the population at that time?

Step-by-Step Solution

Answer

According to the projections, Greece and Belgium will have the same population of 10,400,000 in the year 2025.

Step 1: Define the Linear Functions for the Populations

First, formulate two linear equations, one for each country, to represent the annual changes in population. Let the variable \(x\) represent years since 2000. For Greece, its initial population was 10,600,000 and it's decreasing by 28,000 per year. Therefore, the linear function for Greece's population (we'll denote as \(G\)) is \(G(x) = 10600000 - 28000x\). Following the same logic, the initial population of Belgium was 10,200,000 and it's decreasing by 12,000 per year. So, the linear function for Belgium's population (we'll denote as \( B \)) is \( B(x) = 10200000 - 12000x \).

Step 2: Find the Year When the Two Populations are Equal

Next, find the year when \( G(x) = B(x) \), i.e., when the population of the two countries are equal. To do this, set up the equation \( 10600000 - 28000x = 10200000 - 12000x \) and solve for \( x \). This gives the equation \( 16000x = 400000 \) which simplifies to \( x = 25 \). Therefore, 25 years after the year 2000, or in the year 2025, Greece and Belgium will have the same population.

Step 3: Find the Equal Population Amount

Lastly, to find out what that equal population will be, substitute the value of \( x \) into either of the original equations. For example, for Greece: \( G(25) = 10600000 - 28000*25 = 10400000 \). Therefore, both countries will have a population of 10,400,000 in the year 2025.