Problem 69 - College Algebra and Calculus: An Applied Approach

Question

An airline offers daily flights between Chicago and Denver. The total monthly cost \(C\) (in millions of dollars) of these flights is modeled by \(C=\sqrt{0.25 x+1}\) where \(x\) is the number of passengers flying that month in thousands (see figure). The total cost of the flights for a month is \(3.5\) million dollars. Use the model to determine how many passengers flew that month.

Step-by-Step Solution

Answer

The number of passengers that flew during that month is 45,000.

Step 1: Understand the given equation

The given equation shows that the cost \(C\) (in millions of dollars) of flights is modeled by \(C=\sqrt{0.25x+1}\), where \(x\) is the number of passengers (in thousands). We are given that for a specific month, \(C=3.5\). We have to find the value of \(x\).

Step 2: Solve for x

To solve for \(x\), first square both sides of the equation to get rid of the square root on the right side. \((C)^2 = (3.5)^2 = 0.25x + 1 \) or, \(12.25 = 0.25x + 1\). Next, subtract 1 from both sides, \((12.25 - 1) = 0.25x\) or, \(11.25 = 0.25x\). Lastly, divide both sides by 0.25 to get \(x\). After simplifying, \(x = 45\).

Step 3: Interpret the result

Since \(x\) represents the number of passengers in thousands, this means that 45,000 passengers flew during that specific month.

Key Concepts

Cost Functions

In college algebra, cost functions are essential in understanding how expenses change concerning different variables such as production or output levels. A cost function, typically denoted as \( C(x) \), relates the cost \( C \) to the number of items \( x \) produced or sold. The primary idea is to model and predict expenses using mathematical formulations, allowing businesses like airlines to make informed financial decisions. In our exercise, the total monthly cost \( C \) of flights relates to the number of passengers \( x \) by the equation \( C = \sqrt{0.25x + 1} \).A few things to remember about cost functions:

• They can vary significantly based on the industry and type of operation.
• They are often derived from empirical data, reflecting real-world business expenses.
• This function can include fixed costs (unrelated to the number of units produced) and variable costs (which change with production levels).

In our scenario, even though the cost is defined through a square root equation, it links to the number of passengers flying in a given month, showing how variables influence total costs.

Square Root Equations

Square root equations involve expressions where the variable is within the radical sign. Solving these equations may seem tricky at first, but they follow a systematic process. Let's take a closer look at the equation \( C = \sqrt{0.25x + 1} \) used in the exercise.To solve it, you need to eliminate the square root by squaring both sides. Here's the process explained:

• Square both sides to remove the square root: \( C^2 = 0.25x + 1 \).
• Substitute known values as needed, such as \( C = 3.5 \), to simplify calculations.
• Isolate the variable \( x \) by performing algebraic operations.

Understanding square root equations is crucial because they appear frequently in various mathematical contexts, such as physics and finance. Solving them means getting comfortable with manipulating equations and exponents, helping to build your overall algebra skills.

Problem Solving Involving Equations

Approaching a problem involving equations requires a clear and methodical strategy. Let's break down how these steps help solve our airline passenger problem efficiently:1. **Identify:** Recognize what you are solving for — here, the number of passengers \( x \) in thousands.2. **Understand the Equation:** Know the relationship described by the equation \( C = \sqrt{0.25x + 1} \).3. **Substitute Known Values:** Replace known values, like the cost \( C = 3.5 \), to simplify the equation.4. **Manipulate the Equation:** Perform necessary algebraic operations. Remember that isolating \( x \) often involves: - Squaring to remove the square root. - Simplification by subtraction or addition. - Division or multiplication to isolate the variable.5. **Interpret:** Finally, interpret what your result means in context. For example, \( x = 45 \) indicates 45,000 passengers.Using these steps during any equation problem-solving session can help demystify the task at hand. By following a structured approach, you will improve your problem-solving skills, leading to greater confidence when tackling various algebraic challenges.