Problem 96 - College Algebra with Modeling and Visualization

Question

Suppose that a construction zone can allow 50 cars per hour to pass through and that cars arrive randomly at a rate of \(x\) cars per hour. Then the average number of cars waiting in line to get through the construction zone can be estimated by $$ N(x)=\frac{x^{2}}{2500-50 x} $$ (a) Evaluate \(N(20), N(40),\) and \(N(49)\) (b) Explain what happens to the length of the line as \(x\) approaches \(50 .\) (c) Find any vertical asymptotes of the graph of \(N .\)

Step-by-Step Solution

Answer

(a) \(N(20) \approx 0.267\), \(N(40) = 3.2\), \(N(49) = 48.02\). (b) Lines grow as \(x\) approaches 50. (c) Vertical asymptote at \(x=50\).

Step 1: Evaluating N(20)

To find \(N(20)\), substitute \(x = 20\) into the given function:\[N(20) = \frac{20^2}{2500 - 50 \times 20}\]Calculate the numerator and the denominator:\[N(20) = \frac{400}{2500 - 1000} = \frac{400}{1500}\]Simplify the fraction:\[N(20) = \frac{4}{15}\approx 0.267\]

Step 2: Evaluating N(40)

Substitute \(x = 40\) into the function:\[N(40) = \frac{40^2}{2500 - 50 \times 40}\]Calculate the numerator and the denominator:\[N(40) = \frac{1600}{2500 - 2000} = \frac{1600}{500}\]Simplify the fraction:\[N(40) = \frac{16}{5} = 3.2\]

Step 3: Evaluating N(49)

Substitute \(x = 49\) into the function:\[N(49) = \frac{49^2}{2500 - 50 \times 49}\]Calculate the numerator and the denominator:\[N(49) = \frac{2401}{2500 - 2450} = \frac{2401}{50}\]Evaluate the division:\[N(49) = 48.02\]

Step 4: Behavior as x approaches 50

As \(x\) approaches 50, the denominator \(2500 - 50x\) approaches zero, which makes the value of \(N(x)\) increase significantly towards infinity. This indicates that the number of cars waiting in line grows very large as the rate of arriving cars approaches the zone's maximum capacity of 50 cars per hour.

Step 5: Finding Vertical Asymptotes

A vertical asymptote occurs where the denominator approaches zero, causing the function to become undefined. Set the denominator equal to zero:\[2500 - 50x = 0\]Solve for \(x\):\[50x = 2500\]\[x = 50\]Therefore, there is a vertical asymptote at \(x = 50\).

Key Concepts

Average Number of Cars Waiting

In a construction zone, the average number of cars waiting to get through is a pivotal measure of traffic congestion. The function provided, \( N(x) = \frac{x^2}{2500 - 50x} \), tells us this average based on the car arrival rate \( x \). As cars arrive randomly at the given rate, knowing how many cars on average are stuck in line is useful for planning and managing traffic flow.

• If fewer cars arrive than the zone can handle, fewer cars wait in line.
• If cars arrive near the zone's maximum capacity, more cars wait in line.

This function captures the balance between car arrival rate and maximum passage rate. Calculations from the step-by-step solution show that as \( x \) approaches the limit of 50 cars per hour, the number of waiting cars grows. In practical terms, this could mean longer travel times for drivers. Ensuring that \( x \) remains below the maximum helps reduce waiting times and maintain a smooth traffic flow.

Vertical Asymptotes

Vertical asymptotes are those invisible lines where a graph shoots off towards infinity. These often make the behavior of certain functions quite dramatic. In the context of our construction zone scenario, the function \( N(x) = \frac{x^2}{2500 - 50x} \) reaches a vertical asymptote at \( x = 50 \), as seen in the calculations.

• At a vertical asymptote, the function becomes undefined.
• This stems from the denominator approaching zero, causing \( N(x) \) to soar towards infinity.

For this situation, it signifies a theoretical scenario where an infinite number of cars would be waiting if arrival matched the construction zone's limit. Hence, in practical terms, planners need to be cautious about reaching or exceeding this rate as it can lead to severe traffic congestion.

Limit Behavior in Functions

The concept of limit behavior describes how a function behaves as it approaches a particular point or value. In relation to our construction zone function \( N(x) = \frac{x^2}{2500 - 50x} \), limit behavior is analyzed as \( x \) nears 50.

• As the arrival rate approaches 50, \( N(x) \) climbs higher and higher, indicating longer waits.
• This increase is due to the shrinking denominator ∝ as \( x \) approaches the asymptote value, making \( N(x) \) infinitely large.

Understanding limit behavior is critical for anticipating performance issues in traffic systems. Before reaching \( x = 50 \), actions such as diverting traffic or adjusting the flow can prevent waiting times from becoming unsustainable. Thus, grasping limit behavior can be key to enhancing traffic management strategies.