Problem 93 - College Algebra with Modeling and Visualization

Question

If two parking attendants can wait on 8 vehicles per minute and vehicles are leaving the parking garage randomly at an average rate of \(x\) vehicles per minute, then the average time \(T\) in minutes spent waiting in line and paying the attendant is given by the formula \(T(x)=-\frac{1}{x-8},\) where \(0 \leq x<8 .\) A graph of \(T\) is shown in the figure. (a) Evaluate \(T(4)\) and \(T(7.5) .\) Interpret the results. (b) What happens to the wait as vehicles arrive at an average rate that approaches 8 cars per minute?

Step-by-Step Solution

Answer

At 4 cars/minute, wait is 0.25 minutes; at 7.5 cars/minute, wait is 2 minutes. As the rate approaches 8, the wait time approaches infinity.

Step 1: Understanding the Function

The problem gives us the function for average wait time in line as \( T(x) = -\frac{1}{x-8} \). This function describes the time spent waiting for service as a function of the number of vehicles \( x \) entering the parking garage per minute. Keep in mind that \( x \) must be less than 8.

Step 2: Evaluating T(4)

Substitute \( x = 4 \) into the function: \[ T(4) = -\frac{1}{4-8} = -\frac{1}{-4} = \frac{1}{4} = 0.25 \]This means at an average of 4 vehicles per minute, the wait time is 0.25 minutes.

Step 3: Evaluating T(7.5)

Substitute \( x = 7.5 \) into the function: \[ T(7.5) = -\frac{1}{7.5-8} = -\frac{1}{-0.5} = 2 \]This means at an average rate of 7.5 vehicles per minute, the wait time is 2 minutes.

Step 4: Interpretation of Results

For \( T(4) = 0.25 \), the average wait time is relatively small, indicating that the garage attendants can handle 4 vehicles per minute well. For \( T(7.5) = 2 \), the wait time is significantly longer, showing that as the arrival rate gets closer to 8 vehicles per minute, the waiting time increases.

Step 5: Analysis as x Approaches 8

As \( x \) approaches 8, \( x-8 \) approaches zero, leading \( T(x) = -\frac{1}{x-8} \) to become undefined and approach infinity. This means the wait time becomes extremely long as the arrival rate of vehicles approaches the capacity of the attendants.

Key Concepts

Parking Garage

Managing a parking garage involves not only providing space for vehicles but also efficiently handling the flow of cars arriving and leaving. Parking garages often have attendants who facilitate this process by helping with parking and payment. In our case, imagine a scenario with two attendants responsible for managing incoming vehicles. They can serve up to 8 vehicles per minute. The arrival and service rates directly impact the waiting time and overall efficiency.

• Efficient management minimizes waiting time.
• Understanding the arrival-rate is crucial for planning.
• Proper allocation of resources can enhance service capacity.

Parking garages must strive to balance the number of arrivals with the service capacity to ensure a smooth operation and avoid congestion. This balancing act is vital to maintain a positive user experience.

Function Evaluation

In this context, function evaluation is about applying a specific mathematical formula to assess how the average waiting time changes as the vehicle arrival rate changes. The given function is:\[ T(x) = -\frac{1}{x-8} \]
This function calculates the average waiting time, expressed in minutes, based on the number of vehicles arriving per minute (denoted as \(x\)). Evaluating this function involves substituting different values of \(x\) to see how the wait time changes.

• For \( x = 4 \), \( T(4) = 0.25 \) minutes.
• For \( x = 7.5 \), \( T(7.5) = 2 \) minutes.

This evaluation can help in predicting and managing the efficiency of parking attendants by indicating how varying levels of vehicle influx affect service time. It's crucial to stay within the function’s limit \(0 \leq x<8\) to ensure valid results.

Vehicle Arrival Rate

The arrival rate of vehicles refers to the number of cars entering the parking garage per minute. This rate significantly influences how long each vehicle will need to wait before being serviced. In our scenario:

• The attendants can service up to 8 vehicles per minute.
• An arrival rate approaching this service limit increases the queue size, leading to longer wait times.
• If arrival rates exceed service capacity, delays become inevitable.

Understanding this rate is key for efficient scheduling and resource allocation. Adjusting attendant work-hours and employing real-time monitoring systems can help in managing deal with fluctuating arrival rates effectively.

Service Capacity

Service capacity is the maximum number of vehicles that attendants can handle in a given timeframe without causing excessive waiting. It represents the practical limit on how quickly services like parking, payment, and exit can be processed. In the given problem:

• The service capacity is 8 vehicles per minute.
• The function \( T(x) = -\frac{1}{x-8} \) becomes undefined as \( x \to 8 \), indicating a breakdown if this limit is reached or exceeded.
• Maintaining operations below this capacity is crucial for efficiency.

Proper service capacity management ensures that the parking garage can handle peak times without causing unacceptable delays for customers. Responding to changes in demand by adjusting resources is essential for sustaining high service quality.