Problem 86 - College Algebra with Modeling and Visualization

Question

If a parking garage attendant can wait on 3 vehicles per minute and vehicles are leaving the ramp at \(x\) vehicles per minute, then the average wait in minutes for a car trying to exit is given by the formula \(f(x)=\frac{1}{3-x^{2}}\) (a) Solve the three-part inequality \(5 \leq \frac{1}{3-x} \leq 10\) (b) Interpret your result from part (a).

Step-by-Step Solution

Answer

Vehicles should leave at 2.8 to 2.9 vehicles per minute for 5-10 min wait.

Step 1: Set the inequality

We start with the given three-part inequality: \( 5 \leq \frac{1}{3-x} \leq 10 \). The task requires solving the double inequality to find the range of \( x \).

Step 2: Solve the left side of the inequality

First, let's solve \( 5 \leq \frac{1}{3-x} \). We start by inverting the inequality and flipping the inequality sign, resulting in \( 3-x \leq \frac{1}{5} \). We then solve for \( x \) by subtracting 3 from both sides and multiplying by -1: \( x \geq 3 - \frac{1}{5} = 2.8 \).

Step 3: Solve the right side of the inequality

Next, solve \( \frac{1}{3-x} \leq 10 \). Again, start by inverting the inequality and flipping the inequality sign: \( 3-x \geq \frac{1}{10} \). Solve for \( x \) by subtracting 3 from both sides and multiplying by -1: \( x \leq 3 - \frac{1}{10} = 2.9 \).

Step 4: Combine the solutions

Combine the solutions from Steps 2 and 3 to find \( x \) satisfies both inequalities. The result is \( 2.8 \leq x \leq 2.9 \).

Step 5: Interpret the solution

The solution \( 2.8 \leq x \leq 2.9 \) indicates that the rate at which vehicles leave the ramp should be between 2.8 and 2.9 vehicles per minute to maintain an average wait time between 5 and 10 minutes for a car trying to exit.

Key Concepts

Inequalities

Inequalities are a fundamental concept in college algebra. They involve mathematical expressions that show the relationship of two values not being equal. For instance, we can have an inequality like \( a < b \), which implies that \( a \) is less than \( b \).
In the context of solving inequalities, the goal is to find the values that satisfy the given condition, such as a solution set or interval for a variable. In our example, we start with a double inequality \( 5 \leq \frac{1}{3-x} \leq 10 \).
Here, we are looking for the values of \( x \) that meet both conditions simultaneously.

• Solving such double inequalities requires treating them as two separate inequalities.
• For each inequality, algebraic manipulation can find a range for the variable.
• We confirm the solutions meet the original conditions by testing values.

The boundaries are crucial as they determine valid solutions. So when dealing with inequalities, always pay attention to whether you need to switch inequality signs, especially when multiplying or dividing by negative numbers.

Rational Functions

Rational functions are a type of function that are defined by a fraction in which both the numerator and the denominator are polynomials. These functions often represent real-world situations where quantities are divided by other quantities, like the average wait time in our parking garage example.
The function given in our problem is \( f(x) = \frac{1}{3-x^{-2}} \), showcasing typical features of rational functions:

• They can have vertical asymptotes where the denominator equals zero, making the function undefined at that point.
• They can also include horizontal asymptotes based on the degrees of the polynomials in the numerator and denominator.

Understanding rational functions requires identifying these key features, as they suggest the behavior of the function. For instance, vertical asymptotes determine potential limitations on \( x \), while horizontal asymptotes can hint at end behavior as \( x \) approaches infinity.

Problem Solving

Problem solving in algebra involves using known concepts like inequalities and rational functions to address specific practical problems. This process requires strategically deciphering the problem, manipulating equations, and interpreting solutions.
In our example, solving the inequality \( 5 \leq \frac{1}{3-x} \leq 10 \) involves various steps:

• Breaking down the double inequality into two individual inequalities.
• Using algebraic techniques, such as inversions and sign changes, to isolate \( x \).
• Combining solutions to address the original bounded condition.

After finding solutions, interpreting these results in context provides insights into practical implications, like ensuring efficiency for vehicles in a parking garage. Effective problem solving rests on translating mathematical results into actionable insights, and applying them to real-life scenarios is a powerful algebra skill.