Problem 76 - A Graphical Approach to College Algebra

Question

Highway Design To allow enough distance for cars to pass on two-lane highways, engineers calculate minimum sight distances between curves and hills. The table at top of the next page shows the minimum sight distance \(y\) in feet for a car traveling at \(x\) mph. (Image can't copy) $$\begin{array}{|c|c|c|c|c|} \hline x \text { (in mph) } & 20 & 30 & 40 & 50 \\ \hline y \text { (in feet) } & 810 & 1090 & 1480 & 1840 \end{array}$$ $$\begin{array}{|c|c|c|c} \hline x \text { (in mph) } & 60 & 65 & 70 \\ \hline y \text { (in feet) } & 2140 & 2310 & 2490 \end{array}$$ (a) Make a scatter diagram of the data. (b) Use the regression feature of a calculator to find the best-fitting linear function for the data. Graph the function with the data. (c) Repeat part (b) for a cubic function. (d) Use both functions from parts (b) and (c) to estimate the minimum sight distance for a car traveling 43 mph. (e) Which function fits the data better?

Step-by-Step Solution

Answer

The cubic function fits the data better, providing a more accurate estimate at 43 mph.

Step 1: Plot the Data Points

Create a scatter plot with the given data points. On the x-axis, represent the speed in mph, and on the y-axis, represent the sight distance in feet. Plot each pair (x,y) from the table on this graph.

Step 2: Find Best-Fitting Linear Function

Using a calculator with linear regression capability, input the given data points. The calculator will output an equation of the form \(y = mx + b\), representing the best-fitting linear function.

Step 3: Plot the Linear Function

Use the equation from Step 2 to plot the linear function on the same graph as the scatter plot. This will visually show how well the linear function fits the data.

Step 4: Find Best-Fitting Cubic Function

Use the calculator's cubic regression feature to find a cubic equation of the form \(y = ax^3 + bx^2 + cx + d\) that best fits the data.

Step 5: Plot the Cubic Function

Plot the cubic function on the same graph as the scatter plot and linear function. This will allow for a visual comparison between the two functions.

Step 6: Estimate Sight Distance at 43 mph (Linear)

Substitute \(x = 43\) into the linear equation found in Step 2 to calculate the estimated minimum sight distance for a car traveling at 43 mph.

Step 7: Estimate Sight Distance at 43 mph (Cubic)

Substitute \(x = 43\) into the cubic equation found in Step 4 to calculate the estimated minimum sight distance for a car traveling at 43 mph.

Step 8: Determine Best Fitting Model

Compare the visual closeness of the data points to the plotted linear and cubic functions. Consider statistical measures (like R-squared) from the regression analysis to determine which function provides a better fit.

Key Concepts

Scatter Plot

A scatter plot is a handy tool for visualizing the relationship between two variables. Here, we graph the speed of a car, measured in miles per hour (mph), on the x-axis. The y-axis shows the minimum sight distance in feet. By placing each data point on this graph, we can easily see trends or patterns in the data.
A scatter plot helps us understand if there's a possible relationship—like more speed might need more sight distance. This visual representation of data is the first step in regression analysis, as it sets the stage for determining the best-fitting function.

Linear Regression

Linear regression finds the straight line, defined by the equation \(y = mx + b\), that best represents the data trend. Here, \(m\) is the slope, telling us how much the sight distance increases as speed increases, and \(b\) is the y-intercept, representing the sight distance when the speed is zero mph.
With a calculator, we input the data points, and it outputs these parameters \(m\) and \(b\). The result is a line that passes as close as possible to all our data points, minimizing errors. This straight-line model is a good first approximation for understanding how speed and sight distance might relate.

Cubic Regression

In some cases, data may not fit well with a straight line, as real-world relationships can be more complex. Cubic regression can be used to account for data trends that have curves by using an equation of the form \(y = ax^3 + bx^2 + cx + d\). This equation includes cubic and quadratic terms that help create a curve that more accurately fits the data points.
A cubic regression requires more calculations, but modern calculators simplify this process. The result is a curve that can capture more complicated relationships in the data, reflecting how sight distance changes at different speeds more accurately than a linear function.

Data Fitting

Data fitting involves choosing a model that describes our specific data as accurately as possible. We look for a function, linear or cubic, that closely follows the patterns shown by the scatter plot. Good data fitting not only visually represents the data points on a graph but also provides equations that allow for predictions—such as estimating sight distance at a speed not listed in the original data.
To determine effectiveness, statistical measures like R-squared are used, showing how closely the chosen function matches the actual data. A higher R-squared value suggests a more accurate fit, helping us understand which model, linear or cubic, better predicts sight distances for different speeds.