Problem 60 - A Graphical Approach to College Algebra

Question

Sales of CRT and LCD Screens In the early 21 st cenJury, LCD monitors were a new technology that replaced -CRT (cathode ray tube) monitors. In \(2002,75\) million CRT -nonitors were sold and only 29 million flat LCD (liquid erystal display) monitors were sold. By \(2006,\) the numbers -vere 45 million for CRT monitors and 88 million for LCD nonitors. (Source: International Data Corporation.) -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

Let's set the sales equal: \(-7.5x + 90 = 14.75x - 0.5\). Solving gives \(x=4.07\), so the year is approximately 2004.

Step 1: Define the Variables

We need to represent the year as a variable, and we will use \(x\) for this purpose. Let's set \(x = 2\) to represent the year 2002 and so on. Our equations will model the number of monitors sold in millions.

Step 2: Set Up the Linear Functions

We have two points for each monitor type sales in terms of \(x\) (year): For CRTs: (2, 75) and (6, 45). For LCDs: (2, 29) and (6, 88).Linear equations take the form \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept. We will find \(m\) and \(b\) for both equations.

Step 3: Calculate Slope for CRT Monitors

The slope \(m\) for CRT monitors is calculated as follows:\[ m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{45 - 75}{6 - 2} = -\frac{30}{4} = -7.5 \]The slope of \(-7.5\) means a decrease in sales of 7.5 million CRT monitors per year.

Step 4: Determine the Y-intercept for CRT Monitors

Substitute one point into the equation \(y = -7.5x + b\). Using point (2, 75):\[ 75 = -7.5(2) + b \]\[ 75 = -15 + b \]\[ b = 90 \]Thus, the function for CRT sales is \(C(x) = -7.5x + 90\).

Step 5: Calculate Slope for LCD Monitors

The slope \(m\) for LCD monitors is calculated as:\[ m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{88 - 29}{6 - 2} = \frac{59}{4} = 14.75 \]The slope of \(14.75\) indicates an increase of 14.75 million LCD monitors per year.

Step 6: Determine the Y-intercept for LCD Monitors

Substitute one point into \(y = 14.75x + b\). Using the point (2, 29):\[ 29 = 14.75(2) + b \]\[ 29 = 29.5 + b \]\[ b = -0.5 \]Thus, the function for LCD sales is \(L(x) = 14.75x - 0.5\).

Step 7: Solve for Equal Sales

To find the year when sales were equal, set \(C(x) = L(x)\):\[ -7.5x + 90 = 14.75x - 0.5 \]Combine and simplify:\[ 90 + 0.5 = 14.75x + 7.5x \]\[ 90.5 = 22.25x \]\[ x = \frac{90.5}{22.25} \approx 4.07 \]This corresponds to the year 2004 since \(x=4\) translates to the year 2004.

Key Concepts

Slope Calculation

When you're working with linear functions, one of the key aspects to understand is how to calculate the slope. The slope represents the rate of change between two variables. In the context of our exercise, it helps describe how sales are increasing or decreasing over time.

To calculate the slope, we use the formula \[ m = \frac{y_2 - y_1}{x_2 - x_1} \]where

• \(y_1\) and \(y_2\) are the sales figures for two different years.
• \(x_1\) and \(x_2\) are the corresponding years used as points.

For CRT monitors, the slope \(-7.5\) means the sales decreased by 7.5 million units per year. Conversely, for LCD monitors, the slope \(14.75\) signifies an increase of 14.75 million units per year.

Understanding the slope is crucial because it gives us insights into the trend over time of each product's sales.

Y-intercept

Another essential component of a linear equation is the y-intercept. This is where the line crosses the y-axis. In terms of our exercise, it tells us the starting point of sales when the year variable \(x\) is zero.

To find the y-intercept, substitute one of the given points (sales data) into the linear equation structure \(y = mx + b\) where:

• \(m\) is the slope.
• \(b\) is the y-intercept.

For CRT monitors, using point \((2, 75)\), substituting this into the equation \(-7.5x + b\) yields \(b = 90\). This implies at year \(x = 0\), or year 2000 in our context, there would have been 90 million CRT monitors sold (theoretically). On the other hand, for LCD monitors, substituting \((2, 29)\) into \(14.75x + b\) results in \(b = -0.5\), indicating a theoretical initial figure.

Y-intercepts provide a starting reference point, essential for plotting the sales trend of a product over the years.

Equations of a Line

Finally, for any linear relationship, formulating the equation of a line is critical. These equations enable us to predict future values and analyze trends.

A general linear equation is crafted as \(y = mx + b\), where:

• \(m\) represents the slope.
• \(b\) represents the y-intercept.

For CRT monitors, the equation becomes \(C(x) = -7.5x + 90\). This indicates a steady decline in CRT monitor sales over the indicated time period.

Conversely, the LCD monitors' function is \(L(x) = 14.75x - 0.5\), reflecting a sharp rise in sales during the same period.

Using these equations, one can calculate when two different data sets will intersect, revealing when CRT and LCD sales will be equal — which was determined to be around 2004. Equations of a line distill the relationship into a simple formula that can be used for insightful forecasts and comparisons.