91Ó°ÊÓ

Traffic signals: The number of seconds \(n\) for the yellow light is critical to safety at a traffic signal. One study \({ }^{6}\) recommends the following formula for setting the time: $$ n=t+0.5 \frac{v}{a}+\frac{w+l}{v} . $$ Here \(t\) is the perception-reaction time of a driver in seconds, \(v\) is the average approach speed in feet per second, \(a\) is the deceleration rate in feet per second per second, \(w\) is the crossing-street width in feet, and \(l\) is the average vehicle length in feet. If we assume that the average perception-reaction time is 1 second, the approach velocity is 40 miles per hour ( \(58.7\) feet per second), the deceleration rate is 15 feet per second per second, and the average vehicle length is 20 feet, then \(n\) can be expressed as the following linear function of crossing-street width: $$ n=3.3+0.017 w $$ a. Under the given assumptions, what is the minimum time the yellow light should be on, no matter what the width of the crossing street? b. If the crossing street for one signal is 10 feet wider than the crossing street for another signal, how should the lengths of the yellow light times compare? c. Calculate \(n(70)\) and explain in practical terms what your answer means. d. What crossing-street width would warrant a 5second yellow light?

Step by step solution

01

Understanding Yellow Light Formula

The formula given for the yellow light duration is \(n = t + 0.5 \frac{v}{a} + \frac{w+l}{v}\), where \(t = 1\), \(v = 58.7\ \text{ft/s}\), \(a = 15\ \text{ft/s}^2\) and \(l = 20\ \text{ft}\). This simplifies to \(n = 3.3 + 0.017w\). This equation represents the duration of the yellow light as a linear function of the street width \(w\).

02

Minimum Yellow Light Duration

The minimum duration of the yellow light corresponds to the case where \(w = 0\). Plugging \(w = 0\) into the linear equation \(n = 3.3 + 0.017w\), we find \(n = 3.3 + 0.017 \cdot 0 = 3.3\). Hence, the minimum yellow light time is 3.3 seconds.

03

Comparing Yellow Light Times for Different Widths

For two crossing streets where one is 10 feet wider than the other, we look at the increase in the yellow light duration. The additional time is \(0.017 \times 10\ = 0.17\) seconds. Thus, the yellow light for the wider street should be 0.17 seconds longer.

04

Calculating Yellow Light Duration for 70-foot Width

By setting \(w = 70\), we calculate \(n(70) = 3.3 + 0.017 \times 70 = 4.49\). Therefore, the yellow light should be on for approximately 4.49 seconds if the street is 70 feet wide.

05

Finding Width for a 5-second Yellow Light

To find the width \(w\) that warrants a 5-second yellow light, solve the equation \(5 = 3.3 + 0.017w\). Subtract 3.3 from both sides to get \(1.7 = 0.017w\). Solving for \(w\), we find \(w = \frac{1.7}{0.017} = 100\ \text{feet}\). Thus, a 100-foot wide crossing street requires a 5-second yellow light.

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Key Concepts

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

Linear Functions

A linear function is a type of equation that creates a straight line when graphed. It is often written in the form \( y = mx + b \), where \( m \) represents the slope of the line and \( b \) is the y-intercept.
These functions are straightforward as each variable change leads to a consistent proportional change in the outcome.

• For our exercise, the linear function is given as \( n = 3.3 + 0.017w \).
• Here, \( n \), the yellow light duration, changes linearly with \( w \), the crossing-street width.

The number 3.3 acts as the constant term, indicating the time when the street width is zero.
The slope, 0.017, illustrates how much the yellow light time increases per foot of street width. This relationship helps in understanding how alterations in street dimensions affect traffic signal settings.

Equation Solving

Equation solving is a mathematical process to find the value of a variable that makes an equation true.
This often involves simplifying expressions and isolating the variable on one side of the equation.
Consider the equation for calculating the yellow light time, \( n = 3.3 + 0.017w \).
We can use this equation to find specific values of \( n \) for different \( w \) values by substituting and solving.
For example, to find the minimum yellow light duration, set \( w = 0 \):

• \( n = 3.3 + 0.017 \times 0 = 3.3 \) seconds

Likewise, if the goal is to determine the crossing-street width \( w \) that results in a 5-second yellow light, solve the equation:

• \( 5 = 3.3 + 0.017w \)
• \( 1.7 = 0.017w \)
• \( w = \frac{1.7}{0.017} = 100 \) feet

Breaking down the steps simplifies the solving process.

Variable Substitution

Variable substitution is a method in algebra where specific values replace variables in an equation.
This is a crucial tool for analyzing how changes to one variable affect the entire expression.
For our traffic signal problem, it helps in predicting the yellow light duration under different conditions.
For instance, if \( w = 70 \) feet, substitute this into the equation \( n = 3.3 + 0.017w \):

• \( n = 3.3 + 0.017 \times 70 \)
• \( n = 4.49 \) seconds

This calculation shows how long the yellow light should last when the street width is 70 feet.
Substitution is also used to understand the effect of a 10-foot increase in width:
If one street is 10 feet wider than another, increase \( n \) by \( 0.017 \times 10 = 0.17 \) seconds.
Through substitution, we easily determine the yellow light setting adjustments needed.