Problem 67 - Algebra 1

Question

You and your family take a summer vacation to Ireland. You discover that the number of Americans visiting Ireland is increasing by \(80,000\) visitors per year. Let \(x\) represent the number of visitors in 1997 . If the number of visitors in 1997 was \(700,000,\) how many visitors were expected in \(2000 ?\) Use unit analysis to check your answer.

Step-by-Step Solution

Answer

It is expected that there would be \(940,000\) visitors in 2000.

Step 1: Understand the growth rate

The number of visitors is increasing by \(80,000\) visitors each year. This increase is constant, so it’s a case of linear growth. That is, the number of visitors each year is \(80,000\) more than the last year.

Step 2: Identify the starting point

The problem tells us the number of visitors in 1997 is \(700,000\). This will be our starting point.

Step 3: Calculate the number of years between 1997 and 2000

We subtract 1997 from 2000 to find there are 3 years between them.

Step 4: Calculate the total increase in visitors

As the number of visitors increases by \(80,000\) each year, and there are 3 years between 1997 and 2000, the total increase in visitors over the 3 years is \(80,000 \times 3 = 240,000\).

Step 5: Add the total increase to the initial number of visitors

We add the total increase in visitors over these 3 years, which is \(240,000\), to the initial number of visitors in 1997, which is \(700,000\). The total expected number of visitors in 2000 is \(700,000 + 240,000 = 940,000\).

Key Concepts

Linear Equations

Linear equations are fundamental in understanding relationships where there is a consistent rate of change. Think of them as a mathematical way to represent a straight line, which reveals how one variable changes in relation to another. In our vacation example, the relationship between the number of years since 1997 (\(x\)) and the number of American visitors to Ireland is represented by a linear equation.

More generally, a linear equation in two variables, like \(x\) and \(y\), can be written in the form \(y = mx + b\). Here, \(m\) is the slope or rate of change, and \(b\) is the \(y\)-intercept, where \(y\) is the value when \(x = 0\). So, for our case, the equation would be \(y = 80,000x + 700,000\) where \(y\) is the expected number of visitors and \(x\) is the number of years after 1997.

These equations are not just abstract concepts; they model real-world phenomena, like predicting tourism growth, which is essential for planning and resource allocation.

Unit Analysis

Unit analysis is a handy tool to convert and manipulate units to make sure your math makes sense in the real world. It involves multiplying by fractions that equal one (also known as conversion factors) to change from one unit to another while leaving the actual value unchanged.

In the context of our problem, we are working with the units 'visitors per year.' When we calculate the expected visitors after three years, we multiply the yearly increase (\(80,000\) visitors/year) by the number of years (\(3\) years).

Why is this correct?

Because when we multiply the two, our year units will cancel out (year/year = 1), leaving us with just the number of visitors, which is exactly what we want. This unit consistency confirms that our calculation aligns with the real-world meaning of the problem.

Constant Rate of Change

A constant rate of change is one of the defining characteristics of linear relationships. It means that for every unit increase in one variable, the other variable increases by a fixed amount.

This predictable pattern allows us to make accurate predictions about future values. For instance, knowing that the number of American visitors to Ireland grows by \(80,000\) each year, we can forecast future tourism without having to see the actual yearly data. The rate of change, in this case, is symbolized by the slope (\(m\) in our linear equation), which remains the same across the entire line.

Understanding this concept is crucial in many fields such as economics, physics, and biology because it helps us identify and anticipate trends. For example, a consistent growth in population, investment returns, or bacterial cultures can be described and analyzed with this concept, enabling better planning and decision-making.