Chapter 1.3, Problem 90E

(a)

To determine

The scatter plot of the data

  

Answer to Problem 90E

The value gets increases towards the x-axis. It is lying in the fourth quadrant.

Explanation of Solution

Given information:

Consider the below table which shows that from the top or mountain road a surveyor takes several horizontal measurements x and several vertical measurements y as shown in the table Where ( x and y are measured in feet)

  

Sketch a scatter plot 0f the data.

Formula used:

Plot the x value in horizontal axis and the y axis in vertical axis.

Calculation:

Scattered plot of above considered table data is shown below

  

Conclusion:

The value gets increases towards the x-axis. It is lying in the fourth quadrant.

(b)

To determine

The line that best fits the data using the straightedge method.

Answer to Problem 90E

The value gets increases towards the x-axis. It is lying in the fourth quadrant and forming the straight line.

Explanation of Solution

Given information:

Consider the below table which shows that from the top or mountain road a surveyor takes several horizontal measurements x and several vertical measurements y as shown in the table Where ( x and y are measured in feet)

  

Use a straightedge to sketch the line that you think well fits the data.

Formula used:

Plot the x value in horizontal axis and the y axis in vertical axis.

Calculation:

Straightedge to the sketch line is shown below

  

Conclusion:

The value gets increases towards the x-axis. It is lying in the fourth quadrant and forming the straight line.

(c)

To determine

The equation for the line using the straight edge method.

Answer to Problem 90E

The equation of line formed is $12y=x-600$

Explanation of Solution

Given information:

Consider the below table which shows that from the top or mountain road a surveyor takes several horizontal measurements x and several vertical measurements y as shown in the table Where ( x and y are measured in feet)

  

Find an equation for the line you sketched in part (b)

Formula used:

  $m=\frac{y_2-y_1}{x_2-x_1}$

The equation of the line slope m passing through the point $\left(x_1,y_1\right)$ is given formula $y-y_1=m\left(x-x_1\right)$

Calculation:

Equation of line according above sketch line will be

The slope m of the non vertical line through $\left(x_1,y_1\right)$ and $\left(x_2,y_2\right)$ is

  $m=\frac{y_2-y_1}{x_2-x_1}$

Here, $x_1\ne x_2$

Hence, use the above formula to find the slope m of the non-vertical line through $\left(300,-25\right)$ and $\left(2100,-175\right)$ where $x_1=300,x_2=2100,y_1=-25\text{and }{y}_2=-175$

  $\begin{array}{l}m=\frac{y_2-y_1}{x_2-x_1}\\ =\frac{-175-\left(-25\right)}{2100-300}\\ =\frac{-175+25}{2100-300}\\ =\frac{15}{180}\\ m=-\frac{1}{2}\end{array}$

Slope of the non vertical line through is $m=-\frac{1}{2}\left(300,-25\right)\text{and }\left(2100,-175\right)$

The equation of the line slope m passing through the point $\left(x_1,y_1\right)$ is given formula $y-y_1=m\left(x-x_1\right)$

So use the above considered point $\left(300,-25\right)$ and slopes $m=-\frac{1}{2}$ to find the particular equations where $x_1=300\text{and}{\text{y}}_1=-25$

  $\begin{array}{l}y-y_1=m\left(x-x_1\right)\\ y-\left(-25\right)=-\frac{1}{12}\left(x-300\right)\\ y+25=-\frac{1}{12}\left(x-300\right)\end{array}$

Multiply both side of the find equation $y+25=-\frac{1}{12}\left(x-300\right)$ by 12

  $\begin{array}{l}12\times \left(y+25\right)=-\frac{1}{12}\times 12\left(x-300\right)\\ 12y+300=x-300\end{array}$

Subtract both side of the find equation $12y+300=x-300$ by 300

  $\begin{array}{l}12y+300-300=x-300-300\\ 12y=x-600\end{array}$

Therefore, the equation of line formed is $12y=x-600$

Conclusion:

The equation of line formed is $12y=x-600$

(d)

To determine

The meaning of the slope of the line.

Answer to Problem 90E

The path of the road is being followed in straight way without any turns in between.

Explanation of Solution

Given information:

Consider the below table which shows that from the top or mountain road a surveyor takes several horizontal measurements x and several vertical measurements y as shown in the table Where (x and y are measured in feet)

  

Interpret the meaning of the slope of the line in part (c) in the context of the problem.

Formula used:

  $m=\frac{y_2-y_1}{x_2-x_1}$

The equation of the line slope m passing through the point $\left(x_1,y_1\right)$ is given formula $y-y_1=m\left(x-x_1\right)$

Calculation:

According to the result of part (c) it concludes that the path of the road is being followed in straight way without any turns in between.

Conclusion:

It forms the straight line path without any bends.

(e)

To determine

The sign state for the road.

Answer to Problem 90E

The sign that state on the road will be downhill grade.

Explanation of Solution

Given information:

Consider the below table which shows that from the top or mountain road a surveyor takes several horizontal measurements x and several vertical measurements y as shown in the table Where (x and y are measured in feet)

  

Formula used:

  $m=\frac{y_2-y_1}{x_2-x_1}$

The equation of the line slope m passing through the point $\left(x_1,y_1\right)$ is given formula $y-y_1=m\left(x-x_1\right)$

Calculation:

The sign that state on the road with a downhill grade that has a slope of $-\frac{1}{12}$ for this problem will be 1% grade.

Conclusion:

The sign that state on the road will be downhill grade.