Question

1. social media a nielsen survey of 3000 american moviegoers aged 12–74 found that 27% of them used social media to chat about movies in 2010. the percentage was 29% in 2011 and 31% in 2012. let t = 0, t = 1, and t = 2 correspond to the years 2010, 2011, and 2012, respectively.

a. explain why the three points ( p_1(0, 27) ), ( p_2(1, 29) ), and ( p_3(2, 31) ) lie on a straight line ( l ).
b. if the trend continued, what was the percentage of moviegoers who used social media to chat about movies in 2015?
c. find an equation of ( l ). then use this equation to find and reconcile the result obtained in part (b).
source: nielsen survey.

1. is there a difference between the statements “the slope of a straight line is zero” and “the slope of a straight line does not exist (is not defined)”? explain your answer.
2. consider the slope - intercept form of a straight line ( y = mx + b ). describe the family of straight lines obtained by keeping

a. the value of ( m ) fixed and allowing the value of ( b ) to vary.
b. the value of ( b ) fixed and allowing the value of ( m ) to vary.

Explanation:

Problem 86

Step1: Calculate slope between $P_1,P_2$

Slope formula: $m=\frac{y_2-y_1}{x_2-x_1}$
$m_{12}=\frac{29-27}{1-0}=2$

Step2: Calculate slope between $P_2,P_3$

$m_{23}=\frac{31-29}{2-1}=2$

Step3: Identify t for 2015

2015 corresponds to $t=2015-2010=5$

Step4: Find line equation (slope-intercept)

Slope $m=2$, y-intercept $b=27$, so $y=2t+27$

Step5: Compute 2015 percentage

Substitute $t=5$: $y=2(5)+27=37$

These statements describe two distinct types of lines. A slope of zero means the line is horizontal (parallel to the x-axis, e.g., $y=3$), where the y-value is constant for all x-values. An undefined slope means the line is vertical (parallel to the y-axis, e.g., $x=2$), where the x-value is constant for all y-values, and the slope formula results in division by zero.

a. When the slope $m$ is fixed and the y-intercept $b$ varies, each line has the same steepness and direction but crosses the y-axis at a different point.
b. When the y-intercept $b$ is fixed and the slope $m$ varies, each line crosses the y-axis at the same point $(0,b)$ but has a different steepness or direction.

Answer:

a. The slopes between all pairs of points are equal ($m=2$), so the points lie on a straight line.
b. $37\%$
c. Equation of $L$: $y=2t+27$. Substituting $t=5$ gives $y=37$, which matches the result in part (b).

---

Problem 87