Problem 64 - Algebra A Combined Function

Question

Solve. See Example 7. The number of McDonald's restaurants worldwide in 2009 was 32,478 . In \(2005,\) there were 31,046 McDonald's restaurants worldwide. Let \(y\) be the number of McDonald's restaurants in the year \(x\) where \(x=0\) represents the year 2005. (Source: McDonald's Corporation) a. Write a linear equation that models the growth in the number of McDonald's restaurants worldwide, in terms of the year \(x\). [Hint: The line must pass through (0,31,046) and \((4,32,478) .\) ] b. Use this information to predict the number of McDonald's restaurants in 2012 .

Step-by-Step Solution

Answer

The number of McDonald's restaurants in 2012 is predicted to be 33,552.

Step 1: Identify the coordinates

Given the problem, the coordinates are \((0, 31046)\) for 2005 and \((4, 32478)\) for 2009, with \(x\) representing the years after 2005.

Step 2: Determine the slope (m)

To find the slope of the line, use the formula \( m = \frac{y_2 - y_1}{x_2 - x_1} \). Substituting the values, we have \( m = \frac{32478 - 31046}{4 - 0} = \frac{1432}{4} = 358 \).

Step 3: Write the linear equation

The linear equation in slope-intercept form is \( y = mx + b \). We already have \( m = 358 \) and point \((0, 31046)\), thus \( b = 31046 \). Therefore, the equation is \( y = 358x + 31046 \).

Step 4: Predict the number of restaurants in 2012

For 2012, \(x = 7\) (since 2012 - 2005 = 7). Substitute \(x = 7\) into the equation: \( y = 358(7) + 31046 \). Calculate it to find \( y = 2506 + 31046 = 33552 \).

Key Concepts

Slope-Intercept Form

Linear equations are often represented in the slope-intercept form, which is written as \( y = mx + b \). In this formula, \( m \) stands for the slope of the line, which indicates how much \( y \) increases for each increase in \( x \). The intercept value \( b \) is the point where the line crosses the y-axis. This form is particularly useful in predicting and understanding relationships between variables in coordinate geometry.
In the exercise about McDonald's restaurant growth, you can see this form in action. By determining the slope \( m \) and the intercept \( b \), we form an equation that describes the relationship between years and the number of restaurants: \( y = 358x + 31046 \). This equation tells us that for every additional year after 2005, the number of restaurants increases by 358, starting from 31,046 in the base year.

• The slope \( m \) tells us the rate of change or growth.
• The intercept \( b \) tells us the initial starting value in the context of the year 2005.

Slope Calculation

Calculating the slope is an essential step in forming a linear equation. The slope \( m \) signifies the steepness or direction of a line on a graph. It's calculated by the formula \( m = \frac{y_2 - y_1}{x_2 - x_1} \), which finds out how much the y-coordinate changes with a unit change in the x-coordinate.
In the given task, we have the points

• \((0, 31046)\) for the year 2005
• \((4, 32478)\) for the year 2009

By plugging these into the formula, we found that \( m = \frac{32478 - 31046}{4 - 0} = 358 \). This result shows each year adds 358 more outlets compared to the previous year, emphasizing the consistent growth rate of McDonald's restaurants worldwide. Slope calculation not only simplifies predictions but also helps in understanding trends in data that follow linear patterns.

Coordinate Geometry

Coordinate geometry combines algebra and geometry to study lines and curves using a coordinate plane. Here, every point is defined by a pair of numerical coordinates, helping visualize problems and solutions in mathematics.
In the given exercise, coordinate geometry helps model the number of McDonald's restaurants over time. The coordinates given were

• \((0, 31046)\): representing the base year, 2005.
• \((4, 32478)\): showing the state in 2009.

Using these coordinates, you can graphically plot the relationship between the years and the number of restaurants. By mapping such points and drawing a line through them, coordinate geometry provides a visual representation of the consistent increase in restaurant numbers.Moreover, once the line or the equation is determined, coordinate geometry allows easy prediction of future values, such as the number of restaurants in 2012 (shown when you plot \( x = 7 \)), reinforcing linear prediction insights from algebra.