Problem 82 - College Algebra

Question

Write a piecewise function that models each cellphone billing plan. Then graph the function. \(\$ 60\) per month buys 450 minutes. Additional time costs \(\$ 0.35\) per minute.

Step-by-Step Solution

Answer

The piecewise function is: \(C(x) = \begin{cases} 60 & \text{for} \ x \leq 450 \ 60 + 0.35(x - 450) & \text{for} \ x > 450 \end{cases}\). The graph consists of a horizontal line segment at \( y = 60 \) from \( x = 0 \) to \( x = 450 \), and then a line segment with a slope of 0.35 continuing from \( x = 450 \).

Step 1: Define the first piece of the function

The first part of the function is for when the number of minutes \( x \) is less than or equal to 450. In this situation, the cost \( C \) is simply \(\$60\), or \( C(x) = 60 \).

Step 2: Define the second piece of the function

The second part of this function is for when \( x \) is greater than 450 minutes. Now, you not only have the base cost of \(\$60\), but every additional minute over 450 costs \(\$0.35\). The cost \( C \) can be calculated using the formula: \( C(x) = 60 + 0.35(x - 450) \). The term \( x - 450 \) reflects the amount of minutes used in excess of the 450 covered in the flat rate.

Step 3: Write out the piecewise function

Now, one can write the function \( C(x) \) as: \(C(x) = \begin{cases} 60 & \text{if} \ x \leq 450 \ 60 + 0.35(x - 450) & \text{if} \ x > 450 \end{cases}\)

Step 4: Graph the function

To graph this function, draw a horizontal line at \( y = 60 \) from \( x = 0 \) to \( x = 450 \). This line represents the fixed cost of \(\$60\) when using 450 minutes or less. For \(x > 450\), draw a line with a slope of 0.35 starting at the point (450, 60). The slope represents the extra cost per minute.

Key Concepts

cellphone billing plan

Cellphone billing plans are often structured to accommodate different usage needs and preferences. They typically consist of a flat rate for a set number of minutes and additional charges for extra usage. In this exercise, the plan charges $60 per month, which includes 450 minutes of call time. If a user exceeds 450 minutes, they pay $0.35 for each additional minute.

This structure helps manage the costs for both users and service providers. It allows for predictability and control over expenses, as you would know your bill amount if you stay within the given time limit. Understanding your cellphone billing plan enables informed decisions about your usage, potentially saving money or selecting a plan more suited to your needs.

cost function

A cost function is a mathematical representation that describes how the total cost changes in relation to different variables. In the context of this cellphone billing plan, the function explains how the total cost varies based on the number of minutes used.

Let's break it down:

• For 450 minutes or less, the cost remains constant at \(60. This represents the flat rate part of the billing plan.
• For more than 450 minutes, the additional cost is calculated as \)0.35 per minute. Therefore, the formula becomes \( C(x) = 60 + 0.35(x - 450) \). The term \( x - 450 \) reflects only those minutes which exceed the provided 450 minutes.

Understanding the cost function allows consumers to anticipate their monthly payments accurately, helping them to stay within budget.

graphing functions

Graphing functions provides a visual representation of the relationship between variables. For the piecewise function of this cellphone billing plan, the graph includes two sections:

• A horizontal line at \( y = 60 \) for \( x \leq 450 \) which signifies the fixed cost for up to 450 minutes.
• A line with a positive slope of 0.35 for \( x > 450 \), starting from the point (450, 60), indicating the additional charge per minute beyond the initial 450 minutes. The slope is derived from each additional minute costing $0.35.

Creating such a graph requires plotting these lines correctly based on the given formulas and understanding their points of intersection. It communicates the cost variability visually, making it easier to grasp how costs increase with more usage.

step by step solution

To solve and understand piecewise functions like this one, let's walk through a detailed step-by-step solution:

**Step 1:** Identify the first piece of the function. For this billing plan, when usage is \( x \leq 450 \) minutes, the cost \( C(x) \) is $60 because it is a flat rate.

**Step 2:** Determine the second piece of the function for \( x > 450 \) minutes. Now, the cost function becomes \( C(x) = 60 + 0.35(x - 450) \). The \( x - 450 \) term calculates only those minutes over the 450-minute limit, multiplying by 0.35 to get the additional cost.

**Step 3:** Write the complete piecewise function. This combines both parts into one expression: \[ C(x) = \begin{cases} 60 & \text{if} \ x \leq 450 \ 60 + 0.35(x - 450) & \text{if} \ x > 450 \end{cases} \]

**Step 4:** Graph the function accurately by:

• Drawing a horizontal line at \( y = 60 \) up to \( x = 450 \).
• Then, from point (450, 60), draw a line with slope 0.35 reflecting the additional charge per extra minute.

This thorough breakdown helps solidify understanding of piecewise functions and how they graphically interpret billing plans.