Problem 77 - College Algebra with Modeling and Visualization

Question

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 can be computed by \(f(x)=\frac{x-5}{x^{2}-10 x},\) where \(x>10 .\) (Source: E.Mannering.) (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

(a) \( x \approx 13.9 \); (b) 3 attendants are needed.

Step 1: Understand the Relationship

The function given is \( f(x) = \frac{x-5}{x^2-10x} \), which describes the average waiting time in minutes for each vehicle, where \( x \) is the average number of vehicles being admitted per minute. The task is to find \( x \) such that the waiting time is 15 seconds.

Step 2: Convert Time Units

Since the waiting time provided by the function is in minutes, we need to convert 15 seconds to minutes. 15 seconds is equal to \( \frac{15}{60} = 0.25 \) minutes.

Step 3: Set Up the Equation

To find the admittance rate \( x \), set the function equal to 0.25 minutes: \[ \frac{x-5}{x^2-10x} = 0.25 \]

Step 4: Solve the Equation

Multiply both sides by \( x^2-10x \) to clear the fraction: \( x-5 = 0.25(x^2-10x) \).Expand and simplify: \[ x - 5 = 0.25x^2 - 2.5x \]Rearrange to form a quadratic equation: \( 0.25x^2 - 3.5x + 5 = 0 \).

Step 5: Use the Quadratic Formula

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

Step 6: Calculate Discriminant

Calculate the discriminant \( b^2-4ac \): \( (-3.5)^2 - 4 \cdot 0.25 \cdot 5 = 12.25 - 5 = 7.25 \).

Step 7: Solve for x

Using the quadratic formula: \( x = \frac{-(-3.5) \pm \sqrt{7.25}}{2 \cdot 0.25} \) resulting in two possible values for \( x \), only consider the value where \( x > 10 \). Calculating gives \( x \approx 13.9 \).

Step 8: Evaluate Number of Attendants

Each attendant serves 5 vehicles per minute. Divide the required admittance rate by the service rate of one attendant: \( \frac{13.9}{5} \). Round up, since you cannot have a fraction of an attendant.

Key Concepts

Quadratic Equation

A quadratic equation is a special type of polynomial equation of degree 2. It takes the form \( ax^2 + bx + c = 0 \), where \( a \), \( b \), and \( c \) are constants, and \( a eq 0 \). These equations are fundamental in algebra due to their wide range of applications, including solving real-world problems like the vehicle waiting time at amusement parks.

To solve quadratic equations, we often use the quadratic formula: \( x = \frac{-b \pm \sqrt{b^2-4ac}}{2a} \). This formula helps us find the values of \( x \) that satisfy the equation. In our exercise, we derive a quadratic equation by rearranging a given expression for the average waiting time: \[ 0.25x^2 - 3.5x + 5 = 0 \].

Using this quadratic formula involves several steps:

• Identify \( a \), \( b \), and \( c \) from the equation, which in our example are 0.25, -3.5, and 5 respectively.
• Calculate the discriminant: \( b^2 - 4ac \). A positive discriminant indicates two possible solutions.
• Substitute these values into the quadratic formula to solve for \( x \).

Understanding and applying the quadratic formula allows us to solve for \( x \) effectively, determining the necessary admittance rate to ensure vehicles have minimal waiting time.

Time Conversion

Time conversion is a crucial skill, especially when dealing with equations that involve different time units. In many mathematical problems, consistent units are necessary for accurate calculations. In our given exercise, we need to convert 15 seconds of waiting time into minutes, since the function \( f(x) = \frac{x-5}{x^2-10x} \) provides waiting time in minutes.

To convert seconds into minutes, remember that 60 seconds make up one minute. Thus, to convert seconds to minutes, divide the number of seconds by 60:

• For 15 seconds: \( \frac{15}{60} = 0.25 \) minutes.

This simple conversion step ensures we are comparing and calculating with unified units, which is paramount in any mathematical analysis. Using consistent units helps prevent errors and makes the solution process more straightforward.

Waiting Time Calculation

Calculating waiting time can be an essential part of managing queues and ensuring efficient service in areas like amusement parks or similar settings. In this problem, we calculate how to manage the admittance rate at a gate to maintain a desired waiting time of 15 seconds or less.

Firstly, understanding the provided function \( f(x) = \frac{x-5}{x^2-10x} \) is crucial. It expresses the average waiting time in minutes, dependent on the number of vehicles admitted per minute, \( x \). By solving for \( x \) when the waiting time is 0.25 minutes, we ensure the conditions are met by setting the equation: \[ \frac{x-5}{x^2-10x} = 0.25 \].

Once calculated, we find out the required \( x \) value exceeding 10 vehicles per minute. Furthermore, to determine how many attendants are necessary, knowing one attendant serves 5 vehicles per minute becomes essential. Calculate how many attendants are required by dividing the necessary admittance rate by the rate one attendant handles:

• Admittance rate needed is approximately 13.9 vehicles per minute.
• Thus, \( \frac{13.9}{5} \approx 2.78 \), rounded up suggests 3 attendants are necessary.

Planning the number of attendants ensures the waiting time is capped, optimizing service efficiency and enhancing customer experience.