Problem 115 - College Algebra with Modeling and Visualization

Question

Sales of CRT and LCD Screens In \(2002,75\) million CRT (cathode ray tabe) monitors were sold and 29 million flat LCD (liquid crystal display) monitors were sold. In 2006 the numbers were 45 million for CRT monitors and 88 million for LCD monitors. (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) Interpret the slopes of the graphs of \(C\) and of \(L\) (c) Determine graphically the year when sales of these two types of monitors were equal. (d) Solve part (c) symbolically. (e) Solve part (c) numerically.

Step-by-Step Solution

Answer

Equal sales occurred around mid-2004.

Step 1: Define Variables and Timelines

For CRT and LCD monitors, define the year 2002 as \(x = 0\) and 2006 as \(x = 4\). The sales of CRT monitors in these years are \(75\) million and \(45\) million respectively. LCD sales were \(29\) million and \(88\) million respectively.

Step 2: Construct the Linear Function for CRT Monitors

To find the linear function \(C(x)\) for CRT sales, calculate the slope using the points \((0, 75)\) and \((4, 45)\). The slope \(m\) is \(\frac{45 - 75}{4 - 0} = -7.5\). Hence, the function is \(C(x) = -7.5x + 75\).

Step 3: Construct the Linear Function for LCD Monitors

For LCD monitors, use the points \((0, 29)\) and \((4, 88)\) to find the slope \(m = \frac{88 - 29}{4 - 0} = 14.75\). The linear function for LCDs is \(L(x) = 14.75x + 29\).

Step 4: Interpret the Slopes

The slope \(-7.5\) for CRT indicates a yearly decrease of 7.5 million units in sales. The slope \(14.75\) for LCD suggests a yearly increase of 14.75 million units in sales.

Step 5: Determine Graphically the Year of Equal Sales

Plot \(C(x) = -7.5x + 75\) and \(L(x) = 14.75x + 29\) on a graph. Identify the year where both lines intersect – this represents equal sales of CRT and LCD monitors.

Step 6: Solve Symbolically for Equal Sales

To find when CRT and LCD sales were equal, solve the equation \(-7.5x + 75 = 14.75x + 29\). Rearrange to get \(-22.25x = -46\), thus \(x = 2.067\), which means the sales are equal in a little over 2 years from 2002, around 2004.

Step 7: Solve Numerically for Equal Sales

Using a calculator, verify \(x = 2.067\) by plugging it back into both \(C(x)\) and \(L(x)\) to confirm they are approximately equal. This confirms the sales equality occurred around mid-2004.

Key Concepts

Slope Interpretation

In linear functions, the slope represents the rate of change. Let's break it down:
For CRT monitors, the slope is \(-7.5\), indicating that every year, sales decrease by 7.5 million units. This negative slope shows a decline, suggesting CRT monitors became less popular over the years.
For LCD monitors, the slope is \(14.75\). This positive number means sales increase by 14.75 million units each year. The upward trend indicates LCDs were gaining popularity.

• Negative slope = Decrease over time
• Positive slope = Increase over time

Understanding slope helps predict trends and make informed decisions based on data.

Graphical Solutions

Graphical methods offer visual insights into data relationships. For the monitor sales, we plot two lines:
One line for CRT ( \(C(x) = -7.5x + 75\)) and one for LCD ( \(L(x) = 14.75x + 29\)).
The intersection point of these lines is crucial. It tells us when sales of CRT and LCD monitors were equal. By viewing the graph, you can quickly see that the lines cross a little after two years on the x-axis, indicating equal sales around the year 2004.
This visual approach helps confirm symbolic and numerical solutions.

• Intersection = Equal sales
• Quick, visual understanding of trends

Symbolic Solutions

For a more precise solution, solving symbolically is key. We set the two linear equations equal to find the exact year of equal sales:
\(-7.5x + 75 = 14.75x + 29\)
Rearranging gives us:
\(-22.25x = -46\)
Solving for \(x\), we find \(x \approx 2.067\). This means sales were equal a little over two years from 2002, pinpointing a specific time in 2004.
Using algebraic manipulation can provide the exact crossover point, useful in subjects requiring accuracy like mathematics and economics.

Numerical Methods

Numerical methods involve calculations to verify explicit results. In our case, using a calculator to plug \(x = 2.067\) into both functions:
\(C(x) = -7.5 \times 2.067 + 75\)
and \(L(x) = 14.75 \times 2.067 + 29\),
yields results that confirm the sales figures are about the same. This numerical verification supports earlier findings, solidifying that sales equality happened around mid-2004.
Numerical methods are vital in approximations when precision is critical in real-world problem-solving.