Problem 60 - A Graphical Approach to Precalculus with Limits

Question

Solve each problem. Sales of \(C R T\) and \(L C D\) Screens In the early 21 st century, LCD monitors were a new technology that replaced CRT (cathode ray tube) monitors. In \(2002,75\) million CRT monitors were sold and only 29 million flat LCD (liquid crystal display) monitors were sold. By 2006 the numbers were 45 million for CRT monitors and 88 million for LCD monitors. (a) Find a linear function \(C\) that models these data for CRT monitors and another linear function \(L\) that models these data for LCD monitors. Let \(x\) be the year. (b) Determine the year when sales of these two types of monitors were equal.

Step-by-Step Solution

Answer

In 2004, sales of CRT and LCD monitors were equal.

Step 1: Define Variables

Let \( x \) be the number of years since 2000, which means the year 2002 corresponds to \( x = 2 \), and the year 2006 corresponds to \( x = 6 \). Define \( C(x) \) for CRT monitors and \( L(x) \) for LCD monitors.

Step 2: Determine Linear Model for CRT Monitors

For CRT monitors, we have points (2, 75) and (6, 45). The slope \( m \) is calculated by \( m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{45 - 75}{6 - 2} = -7.5 \). Thus, the linear equation is \( C(x) = -7.5x + b \). Using point (2, 75), find \( b \): \( 75 = -7.5(2) + b \to b = 90 \). Therefore, \( C(x) = -7.5x + 90 \).

Step 3: Determine Linear Model for LCD Monitors

For LCD monitors, we have points (2, 29) and (6, 88). The slope \( m \) is \( m = \frac{88 - 29}{6 - 2} = 14.75 \). The linear equation is \( L(x) = 14.75x + b \). Using point (2, 29), find \( b \): \( 29 = 14.75(2) + b \to b = -0.5 \). Therefore, \( L(x) = 14.75x - 0.5 \).

Step 4: Equate the Functions

Set \( C(x) = L(x) \) to find when CRT and LCD monitor sales were equal. Solve \(-7.5x + 90 = 14.75x - 0.5 \).

Step 5: Solve for x

Rearrange the equation: \(-7.5x + 90 = 14.75x - 0.5 \Rightarrow 90 + 0.5 = 14.75x + 7.5x \Rightarrow 90.5 = 22.25x \Rightarrow x = \frac{90.5}{22.25} \approx 4.07 \).

Step 6: Interpret the Result

The solution \( x \approx 4.07 \) indicates that in the year 2000 + 4.07, or approximately 2004, sales of CRT and LCD monitors were equal.

Key Concepts

Slope

Understanding what a **slope** represents is crucial in analyzing and interpreting data in linear models. The slope of a line in a graph measures the steepness and direction of the line. It is calculated using the formula:\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]It tells us how much the dependent variable (usually on the y-axis) changes for a unit change in the independent variable (usually on the x-axis).

• A positive slope indicates that as x increases, y also increases.
• A negative slope shows that as x increases, y decreases.
• A slope of zero suggests that y does not change as x changes, indicating a horizontal line.

In our example, the CRT monitors have a negative slope of -7.5, indicating declining sales each year, whereas the LCD monitors have a positive slope of 14.75, indicating increasing sales.

Linear Equation

A **linear equation** models a relationship between two variables with constant slope. It typically takes the form:\[ y = mx + b \]where:

• \(m\) is the slope of the line.
• \(b\) is the y-intercept, the value of y when x is zero.

For CRT monitors, the linear equation derived was \( C(x) = -7.5x + 90 \). This indicates that if no time passes (x=0), 90 million monitors are expected to be sold, starting point in absence of other conditions. Similarly, for LCD monitors, the equation was \( L(x) = 14.75x - 0.5 \), which reflects their sales trend over time.
Understanding linear equations helps in predicting trends and interpreting the relationship between variables across various fields.

Equating Functions

**Equating functions** involves setting two functions equal to each other to find when or where they intersect.
In mathematics, this often helps in solving for a specific variable, telling us when two different scenarios yield the same outcome. In our exercise, equating the CRT and LCD sales functions \( C(x) = L(x) \) allows us to find the exact time their sales figures matched.
By solving the equation \(-7.5x + 90 = 14.75x - 0.5\), we calculated x as approximately 4.07. This represents the number of years after 2000 when CRT and LCD sales were equal, concluding it occurred in the year 2004.

Sales Data Analysis

Analyzing **sales data** using linear models provides insights into past trends and can aid in forecasting future outcomes. This approach simplifies complex data interpretation by modeling sales figures with straight lines.
The data from our example, involving CRT and LCD monitor sales over time from a starting year, can be effectively examined using linear functions. Such models allow us to:

• Identify trends over specific periods.
• Predict future sales based on past data.
• Compare and contrast different products or time periods.

By leveraging this analysis, businesses and researchers can make informed decisions regarding production, marketing strategies, and resource allocation.