Math Problem Statement

In 19981998​, the percentage of​ 18- to​ 29-year-olds who used the Internet was 7171​%. In 20072007​, that percentage had risen to 8787​%. Complete parts​ (a) through​ (d) below. Question content area bottom Part 1 ​a) Use the year as the​ x-coordinate and the percentage as the​ y-coordinate. Find the equation of the line that contains the data points. yequals=StartFraction 16 Over 9 EndFraction x minus 3481 16 9x−3481 ​(Type an exact answer. Do not include the​ % symbol in your​ answer.) Part 2 ​b) Use the equation in part​ (a) to estimate the percentage of Internet users in 20122012. The percentage of Internet users in 20122012 is approximately enter your response here​%. ​(Simplify your answer. Use the answer from part​ (a) to find this answer. Round to the nearest tenth as needed. Do not include the​ % symbol in your​ answer.)

Solution

Part 1:

a) Finding the equation of the line:

To find the equation of the line, we will use the two points given, corresponding to the years and percentages:
(1998,71)(1998, 71) and (2007,87)(2007, 87).

The slope mm of the line is calculated as: m=y2y1x2x1=877120071998=169m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{87 - 71}{2007 - 1998} = \frac{16}{9}

Now, using the point-slope form of the equation of a line: yy1=m(xx1)y - y_1 = m(x - x_1) Substitute m=169m = \frac{16}{9} and the point (1998,71)(1998, 71): y71=169(x1998)y - 71 = \frac{16}{9}(x - 1998)

Now, solve for yy: y=169(x1998)+71y = \frac{16}{9}(x - 1998) + 71

Distribute the 169\frac{16}{9}: y=169x169×1998+71y = \frac{16}{9}x - \frac{16}{9} \times 1998 + 71 y=169x3555.11+71y = \frac{16}{9}x - 3555.11 + 71 y=169x3484.11y = \frac{16}{9}x - 3484.11

Thus, the equation of the line is approximately: y=169x3484.11y = \frac{16}{9}x - 3484.11

Part 2:

b) Estimating the percentage of Internet users in 2012:

Now, use the equation y=169x3484.11y = \frac{16}{9}x - 3484.11 and substitute x=2012x = 2012 to estimate the percentage for 2012.

y=169×20123484.11y = \frac{16}{9} \times 2012 - 3484.11 Let me calculate this.The estimated percentage of Internet users in 2012 is approximately 92.8%92.8\%.

Would you like any further clarification or more details?

Here are 5 related questions:

  1. How do you calculate the slope of a line between two points?
  2. What is the point-slope form of a linear equation?
  3. How do you interpret the y-intercept in this context?
  4. How can linear interpolation or extrapolation be used in real-world predictions?
  5. What assumptions are made when predicting future percentages based on past data?

Tip: When solving for a line equation from data points, always verify whether the equation makes sense by substituting values to check consistency.

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Math Problem Analysis

Mathematical Concepts

Linear Equations
Slope-Intercept Form
Interpolation
Extrapolation

Formulas

Slope formula m = (y2 - y1) / (x2 - x1)
Point-slope form y - y1 = m(x - x1)
Equation of a line y = mx + b

Theorems

Slope-Intercept Theorem
Point-Slope Form

Suitable Grade Level

Grades 9-12