Chapter 1.3, Problem 90E

a.

To determine

To verify survey based data.

Answer to Problem 90E

The graph shows the scatter plot of data.

Explanation of Solution

Given information:

From the top of a mountain road, a surveyor takes several horizontal measurements $x$ and several vertical measurements $y$ as shown in the table ( $x$ and $y$ are measured in feet).

  $\begin{array}{cccccccc}x& 300& 600& 900& 1200& 1500& 1800& 2100\\ y& -25& -50& -75& -100& -125& -150& -175\end{array}$

Sketch a scatter plot of the data.

Calculation:

Scatter plot of the data given in table,

  

Hence the graph shows the scatter plot of data.

b.

To determine

To verify survey based data.

Answer to Problem 90E

The graph shows a straightedge.

Explanation of Solution

Given information:

From the top of a mountain road, a surveyor takes several horizontal measurements $x$ and several vertical measurements $y$ as shown in the table ( $x$ and $y$ are measured in feet).

  $\begin{array}{cccccccc}x& 300& 600& 900& 1200& 1500& 1800& 2100\\ y& -25& -50& -75& -100& -125& -150& -175\end{array}$

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

Calculation:

Scatter plot of the data given in table,

  

Hence the graph shows a straightedge.

c.

To determine

To verify survey based data.

Answer to Problem 90E

  $\boxed{12y=x-600}$

Explanation of Solution

Given information:

From the top of a mountain road, a surveyor takes several horizontal measurements $x$ and several vertical measurements $y$ as shown in the table ( $x$ and $y$ are measured in feet).

  $\begin{array}{cccccccc}x& 300& 600& 900& 1200& 1500& 1800& 2100\\ y& -25& -50& -75& -100& -125& -150& -175\end{array}$

Find an equation for the line you sketched in part $\left(b\right)$ .

Calculation:

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

  $m=\frac{y_2-y_1}{x_2-x_1}$

Now find slope $m$ through points $\left(300,0-25\right)\&\left(2100,-175\right)$

  $\begin{array}{l}where\\ x_1=300,x_2=2100\\ y_1=-25,y_2=-175\end{array}$

  $\begin{array}{l}m=\frac{y_2-y_1}{x_2-x_1}\\ m=\frac{-175-\left(-25\right)}{2100-300}\\ =\frac{-175+25}{1800}\\ =\frac{-150}{1800}\\ m=-\frac{1}{12}\end{array}$

Now, with the use of point $\left(300,0-25\right)$ and slope $m=-\frac{1}{12}$ find equation, where $x_1=300,y_1=-25$

  $y-y_1=m\left(x-x_1\right)$

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

Now multiply both side of equation by $12$ ,

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

Now subtract $300$ from both sides of equation,

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

Hence equation of line is obtained $\boxed{12y=x-600}$

d.

To determine

To verify survey based data.

Answer to Problem 90E

The road follows a straight path.

Explanation of Solution

Given information:

From the top of a mountain road, a surveyor takes several horizontal measurements $x$ and several vertical measurements $y$ as shown in the table ( $x$ and $y$ are measured in feet).

  $\begin{array}{cccccccc}x& 300& 600& 900& 1200& 1500& 1800& 2100\\ y& -25& -50& -75& -100& -125& -150& -175\end{array}$

Interpret the meaning of the slope of the line in part $\left(c\right)$ in the context of the problem.

Calculation:

According the results in part $\left(c\right)$ road follow a straight path without any turns.

e.

To determine

To verify survey based data.

Answer to Problem 90E

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

Explanation of Solution

Given information:

From the top of a mountain road, a surveyor takes several horizontal measurements $x$ and several vertical measurements $y$ as shown in the table ( $x$ and $y$ are measured in feet).

  $\begin{array}{cccccccc}x& 300& 600& 900& 1200& 1500& 1800& 2100\\ y& -25& -50& -75& -100& -125& -150& -175\end{array}$

The surveyor needs to put up a road sign that indicates the steepness of the road. For instance, a surveyor would put up a sign that states “ $8\%$ grade” on a road with a downhill grade that has a slope of $-\frac{8}{100}$ . What should the sign state for the road in this problem?

Calculation:

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