Solve each problem. Sales of C R T and L C D Screens In the...

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.

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.

Key Concepts

Linear Function Modeling

This concept involves creating a linear equation to represent a relationship between two variables, where the relationship is assumed to be consistent and the rate of change is constant. It is widely used in data analysis to predict and understand trends over time.

Two-Point Form of a Line

This method allows the determination of a linear equation when two distinct data points are given. By calculating the slope between the points and using one of them to find the intercept, a complete model is constructed to capture the relationship.

Slope and Intercept

In any linear function, the slope represents the rate of change and indicates how much the dependent variable changes with respect to the independent variable, while the y-intercept gives the starting value of the dependent variable when the independent variable is zero.

Solving Linear Equations

This key concept involves setting two linear functions equal to each other to find their point of intersection. It is particularly useful to determine when two different trends or conditions yield the same result, as seen in break-even or equilibrium analysis.

Transcript

00:01 For number 60, we're given data of how many of each type of computer monitor were sold.

00:08 So we have the cathode ray 2, which is the older kind, and lcd, which is the newer kind, and how many million were sold in each of these years.

00:18 And we're to find a linear function that relates to these.

00:22 So first i'm going to do the crt.

00:24 I like to do these as a system of equations.

00:27 You could think of that as two data points and find a slope first that works also.

00:37 So my first data point will be 75.

00:42 I'm just going to know that these are in millions.

00:46 So i'm doing my basic linear equation, y equals mx plus b.

00:54 You're going to do it for my two different points.

00:58 So i'll have a system of equations.

01:01 So here i'll have, my y's will be these, my x's will be the year.

01:08 So 75, i'm looking for the slope.

01:13 My other data point is the 2002 plus b.

01:19 So i have two variables.

01:21 But i'll have another equation over here.

01:22 So this is also for the crt.

01:25 So 45 the slope times the 2006 plus b.

01:34 So another equation with the same two variables.

01:38 I'm going to solve this equation for b.

01:40 So b will equal 45 minus m times 2006 or 2006 m.

01:50 So now i'm going to substitute that over here for the b, and i will have 75 equals 2002m plus b, which is 45 minus minus 2006 m.

02:13 And i'll just combine my like terms and solve.

02:16 So i'm going to subtract 45 from this side, and i'll get 30.

02:25 I'll combine these two, and i get negative 4m.

02:30 I'll divide both sides by negative 4, and i get that my slope is negative 7 .5.

02:43 That's not surprising because it's getting less.

02:45 Each year it's getting less.

02:46 This is the older style monitor.

02:48 So a negative 7 .5.

02:50 And now that i know what my slope is, i can come over here and find my intercept.

02:55 So 45 minus 2006 m, and i know m is negative 7 .5, and that would be my intercept.

03:07 So i multiply and then subtract that from 45.

03:11 So for my intercept, i get 15 ,090.

03:17 So my equation for the cathode ray tube is the number monitors sold as a function of the year is negative 7 .5 times the year plus 1590.

03:43 I'm going to do the same thing now for the lcd monitor...

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