Chapter 8.9, Problem 2GP

(a)

To determine

The slope-intercept form equation for the equation of the line of fit with the help of the following graph:

  

Answer to Problem 2GP

The final slope intercept form equation is derived as $y=0.4x+0.8$

Explanation of Solution

Given:

  

Concept Used:

The slope intercept form equation is

  $y=mx+b$

Where;

Slope: $m$

Y-intercept: $\left(0,b\right)$

Calculation:

Consider,

Year since $1964$ : $x$

Times: $y$

There are two points are given in the graph:

  $\begin{array}{l}A=\left(x_1,y_1\right)=\left(36,52\right)\\ B=\left(x_2,y_2\right)=\left(24,53\right)\end{array}$

Now use any two points to calculate the slope intercept from equation:

  $m=\frac{y_2-y_1}{x_2-x_1}=\frac{53-52}{24-36}=\frac{1}{12}$

Now, the slope intercept form equation is written as:

  $\begin{array}{l}y=mx+b\\ y=\frac{1}{12}x+b\end{array}$

Now put the value of $\left(x_1,y_1\right)=\left(36,52\right)$ in the above eq. to get the Y-intercept:

  $\begin{array}{l}y=mx+b\\ 52=\frac{1}{12}\left(36\right)+b\\ b=52-3\\ b=49\end{array}$

So, the final slope intercept form equation is:

  $\begin{array}{l}y=mx+b\\ y=\frac{1}{12}x+49\end{array}$

Conclusion:

Hence, slope-intercept form is derived with the help of the line of fit.

(b)

To determine

The winning time in the men’s $100m$ butterfly event in the year $2020$ with line of fit equation from the first subpart

Answer to Problem 2GP

Explanation of Solution

Given:

  $x=2020$

Concept Used:

The slope intercept form equation is

  $y=mx+b$

Where;

Slope: $m$

Y-intercept: $\left(0,b\right)$

Calculation:

From the first subpart:

The final slope intercept form equation is derived as:

  $\begin{array}{l}y=mx+b\\ y=\frac{1}{12}x+49\end{array}$

Now, to calculate the winning time $y$ , need to put $x=2020$ in above eq.

  $\begin{array}{l}y=\frac{1}{12}x+49\\ y=\frac{1}{12}\left(2020\right)+49\\ y=168.33+49\\ y=217.33\end{array}$

So, for the year $x=2020$ , there are $y=217.33$ second is the winning time for men’s $100m$ butterfly event.

Conclusion:

Hence, wining time in the year $2020$ is $217.33\sec$ .