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Chapter 3: Problem 7

Slowing down in a curve: A study of average driver speed on rural highways by A. Taragin 4 found a linear relationship between average speed \(S\), in miles per hour, and the amount of curvature \(D\), in degrees, of the road. On a straight road \((D=0)\), the average speed was found to be \(46.26\) miles per hour. This was found to decrease by \(0.746\) mile per hour for each additional degree of curvature. a. Find a linear formula relating speed to curvature. b. Express using functional notation the speed for a road with a curvature of 10 degrees, and then calculate that value.

Short Answer

Expert verified
a. The formula is \(S(D) = 46.26 - 0.746D\). b. With 10 degrees of curvature, the speed is 38.8 mph.

Step by step solution

01

Understanding the Linear Relationship

We are given that speed decreases linearly with the degree of curvature in the form of a linear equation. On a straight road (where curvature \(D = 0\)), the average speed \(S\) is 46.26 mph, and it decreases by 0.746 mph per additional degree of curvature.
02

Constructing the Linear Formula

The linear relationship can be expressed as \(S = mD + c\), where \(m\) is the rate of change in speed per degree of curvature, and \(c\) is the intercept when \(D = 0\). Here, \(m = -0.746\) and \(c = 46.26\). Thus, the formula becomes \(S = 46.26 - 0.746D\).
03

Writing the Functional Notation

To express this relationship as a function, we write \(S(D)\) to signify speed as a function of curvature. Thus, the function is \(S(D) = 46.26 - 0.746D\).
04

Calculating the Speed for 10 Degrees Curvature

Substitute \(D = 10\) into the function: \(S(10) = 46.26 - 0.746 \times 10\). Calculate the value: \(0.746 \times 10 = 7.46\). So, \(S(10) = 46.26 - 7.46 = 38.8\) mph.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Functional Notation
Functional notation is a way to express relationships or formulas in mathematics, making it easier to substitute different values and calculate results. In this exercise, functional notation helps us express speed as a function of road curvature.
This means that instead of writing the formula in a general form, like we do with linear equations, we specify that the value of speed changes with curvature. We denote this relationship with the symbol \(S(D)\). The \(D\) in the parentheses stands for degree of curvature, giving clarity that speed \(S\) depends on \(D\).
The formula is written as \(S(D) = 46.26 - 0.746D\), which neatly shows how the speed varies; when you change the degree of curvature \(D\), you can quickly find the corresponding speed \(S\) by putting that number into the equation.
The beauty of functional notation is in its simplicity and efficiency for calculation. You quickly see which parameters affect the outcome, and it prepares you for higher-level concepts in calculus.
Curvature
Curvature refers to the degree to which a road or any curve bends. In practical terms, it's how sharply you need to turn to stay on the road. In the context of this exercise, curvature is crucial because it affects the speed of a vehicle.
A key point here is that as curvature \(D\) increases, the speed \(S\) decreases. This is due to the natural limitations imposed by sharper turns鈥攖hey necessitate slower speeds for safety and maneuverability.
A road with a curvature of \(0\) degrees is perfectly straight, and in this exercise, corresponds to the maximum speed of \(46.26\) mph. For each subsequent degree of curvature added, the speed decreases by \(0.746\) mph, as depicted by the slope \(-0.746\) in the linear equation.
Understanding curvature is important not just for driving safely, but also for designing roads that optimize traffic flow and vehicle efficiency.
Average Speed
Average speed helps us understand how quickly or slowly something is moving, taking into account the whole trip rather than just instantaneous speed.
In the example given, average speed \(S\) is directly related to the curvature \(D\) of the road due to a linearly decreasing relationship. While driving, average speed tells us how fast you are going considering the entirety of the road's curvature. It combines straight parts and curved sections into one meaningful number.
With the given linear equation \(S = 46.26 - 0.746D\), you can calculate the average speed for any degree of curvature. For a curvature of \(10\) degrees, as calculated in the exercise, we use the functional form \(S(10) = 38.8\) mph.
This calculation helps drivers and engineers predict performance, adjust for safety, and improve road designs, all by simply knowing how average speed adjusts to changes in curvature.

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