Chapter 4, Problem 21CR

To determine

To write: an equation of the roasting time and find slope and y-intercept.

Answer to Problem 21CR

The slope is $=\frac{2.5}{12}$

The Equation that models the roasting time as a function of the weightof the turkey is $y=\frac{2.5}{12}x+\frac{4}{3}$

Explanation of Solution

Given:

  

Calculation:

Step 1

Draw a line that appears to fit the data closely. the scatter plot shows a positive correlation.

  

Step 2 Write an equation using two points on the line. Use $\left(8,3\right)and\left(20,5.5\right)$ . Find the slope of the line using the given points,

  $\left(x_1,y_1\right)=\left(8,3\right)and\left(x_2,y_2\right)=\left(20,5.5\right)$ .

Write the slope formula.

  $m=\frac{y_2-y_1}{x_2-x_1}$

Substitute the values.

  $=\frac{5.5-3}{20-8}$

Simplify.

  $=\frac{2.5}{12}$

Step 3 Now write the equation of the line that passes through the given points in the $point-slope$ form using the slope $m=\frac{2.5}{12}$ and the point $\left(8,3\right)$ .

Write the point-slope form.

  $y-y_1=m\left(x-x_1\right)$

Substitute $\frac{2.5}{12}$ for $m$ , $8$ for $x_1$ , and $3$ for $y_1$ .

  $y-\left(3\right)=\frac{2.5}{12}\left[x-\left(8\right)\right]$

  $y-3=\frac{2.5}{12}\left(x-8\right)$

Distribute multiplication by $\frac{2.5}{12}$

  $\begin{array}{l}m=\frac{2.5}{12}\\ y-3=\frac{2.5}{12}x-\left(\frac{2.5}{12}\times 8\right)\\ y-3=\frac{2.5}{12}x-\left(\frac{2.5}{3}\times 2\right)\\ y-3=\frac{2.5}{12}x-\frac{5}{3}\end{array}$

Add $3$ on both sides.

  $\begin{array}{l}y=\frac{2.5}{12}x+\left(-\frac{5}{3}+3\right)\\ y=\frac{2.5}{12}x+\left(\frac{9-5}{3}\right)\end{array}$

Simplify.

  $y=\frac{2.5}{12}x+\left(\frac{4}{3}\right)$

  $\Rightarrow$ The Equation that models the roasting time as a function of the weightof the turkey is $y=\frac{2.5}{12}x+\frac{4}{3}$

The slope of the line is $\frac{2.5}{12}$ . This means that the roasting time will increase as the weight of the turkey increases. The time will increase by about $\frac{2.5}{12}$ hour for each pound.

  $\begin{array}{l}\frac{2.5}{12}hours=\frac{2.5}{12}\times 60\min utes\\ =12.5minutesforeachpound.\end{array}$

  $y-intercept$ of the line of fit has no meaning in this context because there cannot be any roasting time when the weight of the turkey is 0.

Conclusion:

The slope is $=\frac{2.5}{12}$

The Equation that models the roasting time as a function of the weightof the turkey is $y=\frac{2.5}{12}x+\frac{4}{3}$