Problem 75 - A Graphical Approach to Precalculus with Limits

Question

Time Spent in Line Suppose the average number of vehicles arriving at the main gate of an amusement park is equal to 10 per minute, while the average number of vehicles being admitted through the gate per minute is equal to \(x .\) Then the average waiting time in minutes for each vehicle at the gate is given by $$ f(x)=\frac{x-5}{x^{2}-10 x} $$ where \(x>10\). (Source: Mannering, F. and W. Kilareski, Principles of Highway Engineering and Traffic Analysis, 2d. ed., John Wiley and Sons.) (a) Estimate the admittance rate \(x\) that results in an average wait of 15 seconds. (b) If one attendant can serve 5 vehicles per minute, how many attendants are needed to keep the average wait to 15 seconds or less?

Step-by-Step Solution

Answer

The admittance rate should be approximately 12.38 vehicles per minute, requiring at least 3 attendants.

Step 1: Set up the equation for a 15-second wait

The average waiting time is given by \( f(x) = \frac{x-5}{x^2 - 10x} \). Since waiting time in seconds is 15, convert it to minutes: \( \frac{15}{60} = 0.25 \). We need to solve for \( x \) in the equation \( f(x) = 0.25 \).

Step 2: Solve for x

Set the equation \( \frac{x-5}{x^2 - 10x} = 0.25 \). Cross-multiply to obtain \( x - 5 = 0.25(x^2 - 10x) \), leading to \( x - 5 = 0.25x^2 - 2.5x \). Rearrange the terms to form a quadratic equation: \( 0.25x^2 - 3.5x + 5 = 0 \).

Step 3: Use the quadratic formula

The quadratic equation \( 0.25x^2 - 3.5x + 5 = 0 \) can be solved by the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \) where \( a = 0.25 \), \( b = -3.5 \), and \( c = 5 \). Calculate \( x = \frac{3.5 \pm \sqrt{(-3.5)^2 - 4 \times 0.25 \times 5}}{0.5} \).

Step 4: Calculate the discriminant

Calculate the discriminant \( D = b^2 - 4ac = (-3.5)^2 - 4(0.25)(5) = 12.25 - 5 = 7.25 \). Since \( D > 0 \), there are two real solutions for \( x \).

Step 5: Find the value of x

Calculate \( x = \frac{3.5 \pm \sqrt{7.25}}{0.5} \). \( \sqrt{7.25} \approx 2.69 \), so \( x = \frac{3.5 + 2.69}{0.5} \approx 12.38 \) or \( x = \frac{3.5 - 2.69}{0.5} = 1.62 \). Only \( x \approx 12.38 \) is valid as \( x > 10 \).

Step 6: Determine the number of attendants required

If one attendant serves 5 vehicles per minute, and the admittance rate \( x \approx 12.38 \) vehicles per minute is required, then the number of attendants \( n \) needed is \( n = \frac{12.38}{5} \approx 2.48 \). Since the number of attendants must be whole, round up to 3 attendants.

Key Concepts

Average Waiting Time Calculation

The average waiting time of vehicles at the gate of an amusement park is a critical factor that can affect visitor experience. To estimate this, we use a formula that considers the vehicle arrival rate and the admittance rate. The equation given is:

• \( f(x) = \frac{x-5}{x^2 - 10x} \)

Here, \( x \) represents the admittance rate in vehicles per minute. The numerator, \( x-5 \), accounts for the excess of vehicles per minute after a baseline of 5 vehicles, while the denominator, \( x^2-10x \), models the square of the output rate reduced by the product of incoming vehicles. This mathematical model helps estimate the average delay each car experiences.
To find the admittance rate required for a specific waiting time, such as 15 seconds, we first convert the time into minutes \(0.25 \text{ minutes}\). By substituting this into our equation and solving for \( x \), we determine the optimal rate for maintaining a manageable traffic flow at the main gate.

Traffic Flow Analysis

Traffic flow analysis involves understanding the movement of vehicles through a system such as an amusement park gate. This analysis helps determine the efficiency of vehicle processing and identify bottlenecks. The flow depends on both the arrival rate of vehicles and the processing capability of the attendants.

• The average arrival rate is given as 10 vehicles per minute.
• The rate of processing (admittance rate \( x \)) can be adjusted by the number of attendants.

For a well-functioning system, the processing rate needs to properly balance the arrival rate to prevent backups. The goal is to minimize waiting times while ensuring the system runs smoothly.
By analyzing traffic flow and employing the quadratic equation discussed, effective attendance planning can be achieved to ensure vehicles are processed efficiently and waiting times are reduced.

Quadratic Formula Application

The quadratic formula is essential in many calculations, including optimizing traffic flow, as seen in this exercise. To solve for a specific \( x \) value, we applied the formula:

• \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)

Utilizing the coefficients from our quadratic equation \( 0.25x^2 - 3.5x + 5 = 0 \):

• \( a = 0.25 \)
• \( b = -3.5 \)
• \( c = 5 \)

Calculating the discriminant \( D = b^2 - 4ac \) helps determine the nature of the solutions. Here, \( D = 7.25 \), indicates two viable solutions.
Using these calculations, we find that \( x \approx 12.38 \), indicating the necessary processing rate. It dramatically shows that as problems become complex, quadratic equations and their solutions become invaluable tools in systems optimization, allowing smooth and effective operations.