Internet connections: The number of households that are...

Internet connections: The number of households that are hooked up to the Internet (homes that are online) has been increasing steadily in recent years. In $1995,$ approximately 9 million homes were online. By 2001 this figure had climbed to about 51 million. (a) Use the relation (year, homes online) with $t=0$ corresponding to 1995 to find an equation model for the number of homes online. (b) Discuss what the slope indicates in this context. (c) According to this model, in what year did the first homes begin to come online? (d) If the rate of change stays constant, how many households will be on the Internet in $2006 ?$ (e) How many years after 1995 will there be over 100 million households connected? (f) If there are 115 million households connected, what year is it?

Internet connections: The number of households that are hooked up to the Internet (homes that are online) has been increasing steadily in recent years. In $1995,$ approximately 9 million homes were online. By 2001 this figure had climbed to about
51 million.
(a) Use the relation (year, homes online) with $t=0$ corresponding to 1995 to find an equation model for the number of homes online.
(b) Discuss what the slope indicates in this context.
(c) According to this model, in what year did the first homes begin to come online? (d) If the rate of change stays constant, how many households will be on the Internet in $2006 ?$ (e) How many years after 1995 will there be over 100 million households connected? (f) If there are 115 million households connected, what year is it?

Key Concepts

Solving Linear Equations

Solving linear equations involves isolating the variable of interest, such as time, to determine when a particular condition is met (for instance, reaching a specific threshold). This algebraic manipulation is key to answering questions about future projections or historical beginnings within the modeled scenario.

Extrapolation for Prediction

Extrapolation is the process of using the linear model to estimate values outside the measured range. This is useful for predicting future outcomes or inferring when past events occurred, based on the established trend indicated by the linear equation.

Y-intercept and Time Transformation

By shifting the time variable so that a specific year (e.g., the base year) corresponds to t = 0, the y-intercept of the model directly reflects the initial value of the quantity being measured. This transformation simplifies the model and makes interpretation of the starting conditions more straightforward.

Slope Interpretation

The slope in a linear equation represents the constant rate of change of the dependent variable with respect to the independent variable. It quantifies how much the output (such as the number of households online) changes per unit of time, providing insight into the speed and direction of that change.

Linear Modeling

Linear modeling involves using a straight-line equation to represent relationships between two variables. It is used to capture situations where one variable changes at a constant rate relative to another, and is typically expressed in the form y = mx + b, where m is the slope and b is the y-intercept.

Transcript

00:01 We have a problem regarding internet connections.

00:02 We are told that in 1995, there were 9 million people with internet connections in their houses.

00:11 9 mil.

00:13 And then in 2001, there were 51 million.

00:17 51 mil.

00:19 We need to create a linear equation that models this scenario.

00:22 So let's represent these as points.

00:24 The problem recommends that we call the year 1995 as year zero, so that makes the x value for that point zero, and then the y coordinate would just be nine million.

00:36 Since everything is in millions here, to avoid writing the word million a bunch of times, i'm just going to write this in terms of millions.

00:42 So zero, common nine.

00:44 In the year zero, there were nine times one million people with an internet connection.

00:49 Then the year 2001 is six years after 1995, making it have an x value of six, and a y value of 51 million.

00:58 Okay, we need to be a one.

00:59 Make an equation now.

01:00 In order to make this equation, we need the slope.

01:03 We're going to use the slope formula for this.

01:05 This is m equals y2 minus y1 divided by x2 minus x1, where x1 y1 is a point and x2 y2 is another point.

01:18 So let's plug into this.

01:20 We have, for y2, i have chosen 51.

01:24 So we have 51 minus the other y value of 9 divided by x2 is the x value corresponding with 51, that is 6 minus the x value corresponding with 9, which is 0.

01:35 So we have 51 minus 9 divided by 6 minus 0, which comes out to 42 divided by 6, which is just 7.

01:44 So that's our slope, 7.

01:46 Now it's time to put this into an equation.

01:48 We're going to be using point slope form, which has the generic formula, y minus y1 is equal to m times x minus x1 for a point x1.

01:57 1.

01:58 All right, let's plug in what we've got.

01:59 Y minus 9 is equal to m, the slope, which is 7 times x minus x1, which is just 0.

02:07 Rearranging a little, y minus 9 is equal to 7x or y equals 7x plus 9.

02:15 So this equation is now in slope intercept form, and it will describe the number of houses with an internet connection when you plug in a year for x.

02:23 I'd like to take a moment here to talk about slope first before we begin to do anything else.

02:27 Now, slope is very important in any equation, but in these real -life circumstance equations, such as this one, which is describing a real scenario, the slope takes on a new meaning.

02:38 So our slope here is equal to 7, which could be written as 7 over 1.

02:42 And as you may remember from the textbook, the slope can be written as the change in y, delta y, over the change in x, delta x...

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