Chapter 8.9, Problem 2CYP

A.

To determine

To find: An equation in slope intercept form for the line of fit.

Answer to Problem 2CYP

  $y=-\frac{1}{12}x+55$

Explanation of Solution

Given information:

The scatter plot shows the winning Olympic times in the men’s 100-meter butterfly for several years following 1964.

  

Calculation:

From the figure it is observed that the line passes through the co-ordinates $(24,53)$ and $(36,52)$ .

Let, $(x_1,y_1)=(24,53),(x_2,y_2)=(36,52)$

Then, slope of the line,

  $\begin{array}{l}m=\frac{y_2-y_1}{x_2-x_1}\\ \Rightarrow m=\frac{52-53}{36-24}[\text{Using }(x_1,y_1)=(24,53),(x_2,y_2)=(36,52)]\\ \Rightarrow m=-\frac{1}{12}\end{array}$

The equation of line in point slope form is given by,

  $\begin{array}{l}y-y_1=m(x-x_1)\\ \Rightarrow y-53=-\frac{1}{12}(x-24)[\text{Replacing }m=-\frac{1}{12},(x_1,y_1)=(24,53)]\\ \Rightarrow y-53=-\frac{1}{12}x+2[\because a(b-c)=ab-ac]\\ \Rightarrow y=-\frac{1}{12}x+2+53\\ \Rightarrow y=-\frac{1}{12}x+55\end{array}$

Thus, the equation of the line in slope intercept form is $y=-\frac{1}{12}x+55$ .

B.

To determine

To calculate: The winning time in the men’s 100-meter butterfly in 2012.

Answer to Problem 2CYP

51 s

Explanation of Solution

Given information:

The scatter plot shows the winning Olympic times in the men’s 100-meter butterfly for several years following 1964.

  

Calculation:

Number of years between 1964 and 2012 $=2012-1964=48$

The equation of the line of fit is,

  $\begin{array}{l}y=-\frac{1}{12}x+55\\ \Rightarrow y=-\frac{1}{12}(48)+55[\text{Replacing }x=48]\\ \Rightarrow y=-4+55\\ \Rightarrow y=51\end{array}$

Thus, the winning time in the men’s 100-meter butterfly in 2012 should be 51 s.