Problem 25 - Algebra 2

Question

For Exercises 25 and \(26,\) use the following information. Namid is examining the calling card portion of his phone bill. A 4 -minute call at the night rate cost \(\$ 2.65 .\) A 10 -minute call at the night rate cost \(\$ 4.75\) . Write a linear equation to model this situation.

Step-by-Step Solution

Answer

The linear equation is \( y = 0.35t + 1.25 \).

Step 1: Define the Variables

Let \( x \) represent the cost per minute for the night rate calls and \( c \) represent the fixed charge. We will model the cost of calls using these variables.

Step 2: Set Up Equations Based on Given Data

We have two calls with known lengths and costs. The first call is 4 minutes and costs \( 2.65 \), and the second is 10 minutes and costs \( 4.75 \). This gives us two equations: \( 4x + c = 2.65 \) and \( 10x + c = 4.75 \).

Step 3: Eliminate the Fixed Charge

To find the variable \( x \) which represents the cost per minute, subtract the first equation from the second: \( (10x + c) - (4x + c) = 4.75 - 2.65 \). This simplifies to \( 6x = 2.10 \).

Step 4: Solve for the Cost per Minute

Divide both sides by 6 to solve for \( x \): \( x = \frac{2.10}{6} = 0.35 \). Hence, the cost per minute is \$0.35.

Step 5: Find the Fixed Charge

Substitute \( x = 0.35 \) back into one of the original equations to find \( c \). Using \( 4x + c = 2.65 \), substitute \( x \): \( 4(0.35) + c = 2.65 \). Simplify to get \( 1.40 + c = 2.65 \), so \( c = 1.25 \).

Step 6: Write the Linear Equation

The linear equation that models the situation using the variables \( x \) and \( c \) is \( y = 0.35t + 1.25 \), where \( t \) is the number of minutes and \( y \) is the total cost.

Key Concepts

Cost Analysis

Cost analysis is an important concept in understanding how businesses and individuals evaluate the expenses associated with various choices. In the example given, Namid is looking at the cost associated with nighttime phone calls. To perform a cost analysis, one must consider:

• Fixed Costs: These are constant expenses. In our example, we found the fixed cost to be a fee of $1.25.
• Variable Costs: These change with usage levels. For the phone bill, it's the cost per minute of a call, which was $0.35 per minute.

Identifying both fixed and variable costs is essential as it aids in understanding how the total cost accumulates over time. This analysis helps in making informed decisions on budget allocation and cost-saving strategies.
A thorough cost analysis provides clarity and reveals the underlying patterns of spendings, such as whether reducing call times can lead to significant savings.

Problem-Solving

Effective problem-solving involves a structured approach to finding solutions. It starts by clearly understanding the problem at hand. In Namid's case, the challenge was to determine the per-minute rate and fixed charge for his phone calls.
The next step is to gather information, as Namid did when he recorded the length and cost of two different calls. With this information, it's possible to employ techniques such as:

• Setting up equations: Identify variables and the relationships between them.
• Performing operations: Use arithmetic operations strategically to isolate and solve for unknowns.

In Namid's exercise, the subtraction of equations helped to remove unknowns, making the problem simpler to solve. Systematic problem-solving leads to clarity, providing a clear path to the solution. Through practice, these skills can become second nature, helping tackle more complex problems efficiently.

Algebraic Modeling

Algebraic modeling is the process of using algebraic expressions to represent real-world situations. In the example given, Namid seeks to model his phone bill costs in terms of a linear equation. The equation derived can be a powerful tool:

• Represents Relationships: The equation shows how changes in the length of the call impact the total cost.
• Predicts Outcomes: By plugging in different call durations, Namid can predict costs accurately.

The model Namid developed is a linear one, represented by the equation: \[ y = 0.35t + 1.25 \]Here, \( t \) represents the number of minutes, and \( y \) is the resulting total cost.
Linear models are prevalent in financial forecasting, economics, and various scientific fields because they show direct relationships between variables. Mastering algebraic modeling empowers one to simplify complex scenarios and make well-informed predictions.