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Chapter 2: Problem 30

A phone company charges for service according to the formula: \(C(n)=26+0.04 n\), 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.

Short Answer

Expert verified
The rate of change is $0.04 per minute, and the initial value is $26.

Step by step solution

01

Identifying the Formula

The formula given is \(C(n) = 26 + 0.04n\). This is a linear function, which can be written in the form \(y = mx + b\) where \(m\) is the rate of change and \(b\) is the initial value.
02

Finding the Rate of Change

In the equation \(C(n) = 26 + 0.04n\), the rate of change is represented by the coefficient of \(n\), which is \(0.04\). This means that for every one additional minute talked, the cost increases by \($0.04\).
03

Finding the Initial Value

The initial value in the equation \(C(n) = 26 + 0.04n\) is \(26\). This is the cost of the service when no minutes are talked, meaning the fixed charge per month for having the phone service.
04

Interpretation of Rate of Change and Initial Value

The rate of change, \(0.04\), indicates the incremental cost per minute of phone use. The initial value, \(26\), represents the base cost incurred regardless of the number of minutes used.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding Rate of Change
The rate of change in a linear function tells us how one variable changes in relation to another. It is the "slope" in the equation. In the context of the phone company's cost formula, the rate of change is the price per minute of phone use, which is represented by the number 0.04 in the equation \(C(n) = 26 + 0.04n\). This value indicates that for each minute talked, the cost increases by 0.04 dollars or 4 cents. Understanding rate of change helps us see how costs increase with increased usage. Simply put: - The rate of change = 0.04 means the charge goes up by 4 cents per additional minute.The rate of change is crucial for predicting costs and budgeting communications effectively.
Defining Initial Value
The initial value in a function is essentially the starting point, also called the "y-intercept". It tells us the value of the function when the input is zero. In the equation \(C(n) = 26 + 0.04n\), the number 26 represents the initial value. This is the fixed cost associated with the phone service, even if no time is spent talking on the phone.It shows what you will be charged regardless of usage:- Initial Value = 26 dollars, implying a base cost of using the service each month.This fixed charge is helpful for understanding the basic cost without using the service extensively. It's a baseline that customers need to be aware of when signing up for the service.
Interpretation of Functions
Interpreting functions involves understanding what each part of a function tells us about a real-world scenario. For the phone company, the function \(C(n) = 26 + 0.04n\) breaks down into understandable components:- The rate of change (0.04) represents the cost per additional minute.- The initial value (26) shows the base price before usage is factored in.Putting these together, you can interpret this function as: - The monthly cost starts at 26 dollars, and then increases as more minutes are used, at a rate of 0.04 dollars per minute.This interpretation is beneficial for users to predict their monthly expenses based on their talking habits, allowing them to make informed decisions about their phone use and budget.

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