From the top of a mountain road, a surveyor takes several...

From the top of a mountain road, a surveyor takes several horizontal measurements and several vertical measurements as shown in the table ( and are measured in feet). $$ \begin{array}{|c|c|c|c|c|} \hline x & 300 & 600 & 900 & 1200 \\ \hline y & -25 & -50 & -75 & -100 \\ \hline \end{array} $$ $$ \begin{array}{|c|c|c|c|} \hline x & 1500 & 1800 & 2100 \\ \hline y & -125 & -150 & -175 \\ \hline \end{array} $$ (a) Sketch a scatter plot of the data. (b) Use a straightedge to sketch the line that you think best fits the data. (c) Find an equation for the line you sketched in part (b) (d) Interpret the meaning of the slope of the line in part (c) in the context of the problem. (e) 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?

From the top of a mountain road, a surveyor takes several horizontal measurements and several vertical measurements as shown in the table ( and are measured in feet).
$$
\begin{array}{|c|c|c|c|c|}
\hline x & 300 & 600 & 900 & 1200 \\
\hline y & -25 & -50 & -75 & -100 \\
\hline
\end{array}
$$
$$
\begin{array}{|c|c|c|c|}
\hline x & 1500 & 1800 & 2100 \\
\hline y & -125 & -150 & -175 \\
\hline
\end{array}
$$

(a) Sketch a scatter plot of the data.
(b) Use a straightedge to sketch the line that you think best fits the data.
(c) Find an equation for the line you sketched in part (b)
(d) Interpret the meaning of the slope of the line in part (c) in the context of the problem.
(e) 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?

Key Concepts

Interpretation of Slope in Context

Interpreting the slope in context means explaining what the slope value represents in a real-world setting. For example, in a situation involving elevation changes, the slope can explain how much vertical change occurs over a given horizontal distance, directly relating to concepts like steepness.

Percentage Grade

Percentage grade is a practical way to express the steepness of a slope by converting the slope from a decimal to a percentage. This involves multiplying the slope by 100 and is commonly used to communicate the incline or decline of roads and other surfaces.

Slope

The slope of a line indicates how steep the line is by quantifying the rate of change of the dependent variable relative to the independent variable. It is calculated as the vertical change divided by the horizontal change between two points on the line.

Linear Equation

A linear equation describes a straight line in the coordinate plane and is usually expressed in the slope-intercept form, y = mx + b. This form helps to clearly identify the slope (rate of change) and the y-intercept (the value of y when x equals zero).

Line of Best Fit

The line of best fit is a straight line drawn through a scatter plot which represents the overall trend of the data. It minimizes the distances – often in a least squares sense – between the actual data points and the line, providing a simple predictive model.

Scatter Plot

A scatter plot is a type of graph that uses Cartesian coordinates to display values for two variables. Each point represents an observation from the data set, making it useful for assessing the relationship between the independent and dependent variables.

Transcript

00:01 In this problem, we've given some data that has been collected by a road surveyor of horizontal measurements and vertical measurements, and given in the table here, measured in feet.

00:12 The first thing we're asked to do is to sketch a scatter plot of the data.

00:17 Remember that x values go left and right, y values go up and down.

00:21 So we're going to go right 300 and down 25, right 600, and down 50, right 900, right 900 ,000.

00:30 Down 75, right, 1 ,200, down 100, right 1 ,500 down 125, right 1 ,800, right, 1800, down 150, and right 2100 down 175.

00:45 This is a very negative linear correlation.

00:52 Here, this is actually a line, so we have a negative 1r value here.

00:57 We can connect this with a line.

00:59 What we'd want to do is extend this up through the x and y intercepts here because that's going to help us get the equation of this line.

01:10 So here's our line of best fit.

01:12 It actually goes through all of the points.

01:14 And looking for the equation of this line.

01:17 Remember that the equation of the line is y equals mx plus b.