Chapter 8.9, Problem 4PPS

(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 4PPS

The final slope intercept form equation is derived as $y=-17x+416$

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: $x$

Number: $y$

There are two points are given in the graph:

  $\begin{array}{l}A=\left(x_1,y_1\right)=\left(4,348\right)\\ B=\left(x_2,y_2\right)=\left(6,314\right)\end{array}$

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

  $m=\frac{y_2-y_1}{x_2-x_1}=\frac{314-348}{6-4}=\frac{-34}{2}=-17$

Now, the slope intercept form equation is written as:

  $\begin{array}{l}y=mx+b\\ y=-17x+b\end{array}$

Now put the value of $\left(x_1,y_1\right)=\left(6,314\right)$ in the above eq. to get the Y-intercept:

  $\begin{array}{l}y=mx+b\\ 314=-17\left(6\right)+b\\ b=314+102\\ b=416\end{array}$

So, the final slope intercept form equation is:

  $\begin{array}{l}y=mx+b\\ y=-17x+416\end{array}$

Conclusion:

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

(b)

To determine

The number of fans in the year of $2013$ in the US

Answer to Problem 4PPS

The number of fans in the year of $2013$ in the US is $195$

Explanation of Solution

Given:

  $x=13$

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:

  $\begin{array}{l}y=mx+b\\ y=-17x+416\end{array}$

Now, to calculate number of fans $y$ , need to put $x=13$ in above eq.

  $\begin{array}{l}y=mx+b\\ y=-17\left(13\right)+416\\ y=-221+416\\ y=195\end{array}$

So, number of fans in the year of $2013$ in the US is $y=195$

Conclusion:

Hence, number of fans is calculated as $y=195$