Problem 117 - College Algebra

Question

A phone company charges for service according to the formula: \(C(n)=24+0.1 n,\) where \(n\) is the number of minutes talked, and \(C(n)\) is the monthly charge, in dollars. Find and interpret the rate of change and initial value.

Step-by-Step Solution

Answer

The initial value is $24, representing the base monthly fee; the rate of change is 0.1, meaning each minute costs an additional $0.10.

Step 1: Understand the Formula

The given formula is \(C(n) = 24 + 0.1n\). This is a linear equation where \(C(n)\) represents the monthly charge in dollars and \(n\) is the number of minutes talked. We need to interpret the components of this equation: the constant \(24\) and the coefficient \(0.1\).

Step 2: Identify the Initial Value

In the linear equation \(C(n) = 24 + 0.1n\), the initial value is the constant term, \(24\). This represents the fixed monthly charge, or the baseline cost for using the service even if no minutes are used.

Step 3: Determine the Rate of Change

The rate of change in the equation is represented by the coefficient of \(n\), which is \(0.1\). This means that for each additional minute talked, the monthly charge increases by \(0.1\) dollars, or 10 cents.

Step 4: Interpret the Values

The initial value of \(24\) tells us that the monthly service fee, before any minutes are used, is \\(24. The rate of change, \(0.1\), indicates that every additional minute of talk time adds \\)0.10 to the total monthly charge.

Key Concepts

Rate of Change

In the context of the phone company's charging formula, the rate of change refers to how much the total cost increases with each additional minute talked. The formula given is \( C(n) = 24 + 0.1n \), where \( n \) is the number of minutes talked. Here, the rate of change is represented by the coefficient of \( n \), which is \( 0.1 \).This coefficient tells us:

• For every extra minute of conversation, the cost increases by \( 0.1 \) dollars.
• Another way to express this is that each minute adds 10 cents to the total bill.

Understanding the rate of change is crucial because it helps predict how changing the number of minutes affects the overall cost. Breaking down the formula in such a manner simplifies the process for estimating usage costs.

Initial Value

The initial value in a linear equation such as \( C(n) = 24 + 0.1n \) is the starting point of your calculation when \( n = 0 \), indicating no minutes have been used. In this formula, the initial value is represented by the constant term, \( 24 \):

• This is the cost you incur without talking any minutes – the baseline price.
• It represents a fixed charge just to have access to the company's service, irrespective of how much it is used.

The initial value is simple yet important because:- It reveals any non-variable costs you must always pay.- It affects how affordable the service is for potential low-usage customers.Recognizing the initial value helps in planning and managing expectations about your subscription costs.

Linear Equation Interpretation

A linear equation like \( C(n) = 24 + 0.1n \) is a straightforward way to calculate potential costs based on variable usage.Explaining the components:

• \( C(n) \) is the total cost, which is dependent on \( n \), the number of minutes.
• The term \( 24 \) is the non-negotiable starting price (also known as the initial value).
• The term \( 0.1n \) calculates additional costs as minutes increase (the rate of change).

Using this equation, you can:- Easily determine how much more the charge will be if you talk for more minutes.- Plan your phone usage and budget by calculating estimated monthly charges based on your expected minute usage.Recognizing how each part of a linear equation contributes to the overall picture makes it easier to make informed decisions about service usage.