Problem 95 - College Algebra with Modeling and Visualization

Question

Suppose that a parking attendant can wait on 40 cars per hour and that cars arrive randomly at a rate of \(x\) cars per hour. Then the average number of cars waiting in line can be cstimated by $$ N(x)=\frac{x^{2}}{1600-40 x} $$ (a) Evaluate \(N(20)\) and \(N(39)\) (b) Explain what happens to the length of the line as \(x\) approaches 40 (c) Find any vertical asymptotes of the graph of \(N\).

Step-by-Step Solution

Answer

(a) \(N(20) = 0.5\), \(N(39) = 38.025\); (b) Line length approaches infinity as \(x\) approaches 40; (c) Vertical asymptote at \(x = 40\).

Step 1: Evaluate N(20)

To find \(N(20)\), substitute \(x = 20\) into the function \(N(x) = \frac{x^2}{1600 - 40x}\). Calculate:\[N(20) = \frac{20^2}{1600 - 40 \times 20} = \frac{400}{1600 - 800} = \frac{400}{800} = 0.5\]

Step 2: Evaluate N(39)

Substitute \(x = 39\) into the function \(N(x) = \frac{x^2}{1600 - 40x}\). Perform the calculation:\[N(39) = \frac{39^2}{1600 - 40 \times 39} = \frac{1521}{1600 - 1560} = \frac{1521}{40} = 38.025\]

Step 3: Analyze behavior as x approaches 40

As \(x\) approaches 40, examine the denominator of \(N(x) = \frac{x^2}{1600 - 40x}\). The denominator \(1600 - 40x\) approaches zero, causing \(N(x)\) to increase dramatically. Hence, as \(x\) approaches 40, the line length approaches infinity.

Step 4: Find the vertical asymptote

The function \(N(x)\) has a vertical asymptote where the denominator equals zero. Solve \(1600 - 40x = 0\):\[40x = 1600 \x = 40\]Thus, the vertical asymptote is at \(x = 40\).

Key Concepts

Rational Functions

Rational functions are a special type of function which are expressed as the ratio of two polynomials. In mathematical terms, a rational function is represented as \( R(x) = \frac{P(x)}{Q(x)} \), where both \( P(x) \) and \( Q(x) \) are polynomials and \( Q(x) eq 0 \).
In our example, the function \( N(x) = \frac{x^2}{1600 - 40x} \) is a rational function. Here, \( x^2 \) is the polynomial in the numerator and \( 1600 - 40x \) is the polynomial in the denominator.
This type of function is commonly used to model real-world situations, like the number of cars waiting at a parking lot, which involves division of one quantity by another.

• The numerator determines the values based on inputs, \( x \).
• The denominator ensures that the function doesn't allow division by zero, which is crucial in defining the domain.

Understanding rational functions helps in identifying points where the function can become undefined, which leads us to the concept of vertical asymptotes.

Vertical Asymptotes

Vertical asymptotes occur in rational functions where the value of the denominator approaches zero, leading the function value towards infinity. This means the function graph will shoot up or down depending on the sign but never actually touch or cross these vertical lines.
For the function \( N(x) = \frac{x^2}{1600 - 40x} \), to find the vertical asymptote, we need to solve for when the denominator is zero, i.e., \( 1600 - 40x = 0 \). Solving this equation, we get \( x = 40 \).

• At \( x = 40 \), the function has a vertical asymptote.
• Vertical asymptotes help in understanding the behavior of the function as it approaches certain values.

In real-world contexts, like our parking situation, it highlights a point where the system might be overwhelmed or a limit beyond which normal operations can't continue.

Limits in Calculus

Limits are a fundamental concept in calculus used to analyze how functions behave as they approach a specific input value. They help in understanding the behavior of functions near points of discontinuity.
For our function \( N(x) = \frac{x^2}{1600 - 40x} \), as \( x \) approaches 40, the denominator approaches zero, increasing the value of \( N(x) \) towards infinity. This illustrates a key aspect of limits: as the input nears a critical value, the output can often become unbounded.

• Limits allow us to make sense of infinite behaviors in functions without requiring each point to be defined.
• Understanding the limit of \( N(x) \) as \( x \to 40 \) shows why \( N(x) \) becomes infinitely large at this point.

In calculus, finding these limits helps in predicting trends and making informed decisions, such as anticipating excessive queues in a service scenario.