Question

Based on data from \(1991-1995,\) a model for the annual number of visitors \(v\) from the United States to Europe is \(v=559,100 t+6,423,000\) where \(t\) is the number of years since \(1991 .\) According to this model, in what year will the number of visitors to Europe reach \(11,000,000 ?\)

Short answer

As \(t\) is measured in whole years, and \(t = 8.18\) is not a whole number, we need to round \(t\) to the closest whole year, which means the number of visitors will reach 11,000,000 in the year 2000.

Step-by-step solution

Set up the equation

First, set the visitor model equal to the desired number of visitors. So, \(559,100 t+6,423,000 = 11,000,000\)

Solve for \(t\)

Now, rearrange the equation to solve for \(t\). Subtract \(6,423,000\) from both sides of the equation. This results in \(559,100 t = 4,577,000\). Then, divide both sides of the equation by \(559,100\) to solve for \(t\). Thus, \(t = \frac{4,577,000}{559,100}\)

Calculate the value of \(t\)

Carry out the division to get the value of \(t\) in years. \(t = \frac{4,577,000}{559,100} = 8.18\) years.

Find the particular year

The value of \(t\) corresponds to the number of years since 1991. Therefore, to get the specific year, add \(t\) to 1991. Hence, year when the populace will reach 11,000,000 = \(1991 + t = 1991 + 8.18\).

Key concepts

Understanding Linear Equations

Linear equations form the foundation of algebra and represent relationships with a straight line when graphed on a coordinate plane. Their general form is \(ax+b=c\), where \(a\), \(b\), and \(c\) are constants and \(x\) is the variable. In practical situations, like forecasting the number of visitors based on a specific trend, we use linear equations such as \(v=559,100t+6,423,000\) to predict future values.

When we want to find a specific outcome, for instance, when the visitor count will reach 11 million, we set the linear equation equal to that value to determine the unknown variable, in this case, \(t\). By manipulating this equation, subtracting \(6,423,000\) from both sides, and then dividing by the coefficient in front of \(t\) which is \(559,100\), we isolate the variable and solve the equation.

Exploring Linear Models

Linear models like \(v=559,100t+6,423,000\) use linear equations to represent real-world phenomena. These models can forecast trends over time, such as predicting the number of visitors from the United States to Europe.

They are built on the assumption that the rate of change is constant, which means the increase or decrease over each period is the same. The equation given represents a linear relationship between the annual number of visitors \(v\) and the number of years since 1991, denoted by \(t\). The coefficient of \(t\), in this case, \(559,100\), signifies the annual increase in visitors, and \(6,423,000\) represents the initial number of visitors in the base year. By solving a linear model for a given value, we can predict significant milestones or intersections.

Algebraic Expressions in Real Life

Algebraic expressions are mathematical phrases that contain numbers, variables like \(t\), and operations but do not include an equality sign. In the given linear model \(v=559,100t+6,423,000\), \(559,100t\) and \(6,423,000\) are algebraic expressions representing the change in visitor numbers and the base number of visitors, respectively.

An important aspect of algebraic expressions is the use of variables to represent unknowns or quantities that can change. By solving the linear equation derived from these expressions, we can uncover valuable insights into trends and make informed decisions or predictions in various fields such as economics, science, and engineering.