Problem 32 - College Algebra Essentials

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 around the year 2025 (2000 + 25) and the population will be approximately 10,200,000 at that time.

Step 1: Set up the Equations

We set up the equations representing the time-dependent populations of Greece and Belgium respectively. Let the year 2000 be represented by t = 0. The population of Greece decreases by 28,000 people each year, so the equation is P_greece(t) = 10,600,000 - 28,000 * t. For Belgium, the population decreases by 12,000 each year, thus P_belgium(t) = 10,200,000 - 12,000 * t.

Step 2: Equate the Two Equations

The next step involves setting the two equations equal to each other, because the two countries will have the same population when their population equations intersect. So, we set them equal: 10,600,000 - 28,000t = 10,200,000 - 12,000t.

Step 3: Solve for t

Solving the equation above for t, we first subtract 10,200,000 from both sides, and then combine like terms to isolate t. The result is 16,000t = 400,000. Finally we divide both sides by 16,000 to get t ≈ 25.

Step 4: Find the Population

With the time, t, when both populations will be the same known, substitute t = 25 into either of the original equations to find the population at that time. For instance, substituting t = 25 into the equation for Greece's population gives : P_greece(25) = 10,600,000 - 28,000 * 25 ≈10,200,000.