Problem 59 - A Graphical Approach to Precalculus with Limits

Question

Suppose that the charges for an international cellular phone call are \(\$ 0.50\) for the first minute and \(\$ 0.25\) for each additional minute. Assume that a fraction of a minute is rounded up. (a) Determine the cost of a phone call lasting 3.5 minutes. (b) Find a formula for a function \(f\) that computes the cost of a telephone call \(x\) minutes long, where \(0< x \leq 5\) (Hint: Express \(f\) as a piece wise- defined function.)

Step-by-Step Solution

Answer

(a) $1.25 for a 3.5 minute call. (b) \(f(x) = \begin{cases} 0.50, & 0< x \leq 1 \\ 0.50 + 0.25 \times \lceil x - 1 \rceil, & 1 < x \leq 5 \end{cases}\).

Step 1: Understanding the Cost of a 3.5 Minute Call

The cost structure specifies $0.50 for the first minute and $0.25 for each additional minute. First, calculate the cost for the first minute: $0.50. Then, calculate the remaining time, which is 3.5 - 1 = 2.5 minutes. Since fractions of a minute are rounded up, we count this as 3 additional minutes. The cost for these is 3 * $0.25 = $0.75. Therefore, the total cost is $0.50 + $0.75 = $1.25.

Step 2: Creating a Function for Call Costs Up to 5 Minutes

Given the constraints, the function will depend on whether the call is less than or more than one minute. For calls less than or equal to a minute: if \(0 < x \leq 1\), then \(f(x) = 0.50\). For calls longer than a minute: if \(1 < x \leq 5\), we see that the remaining time, \(x - 1\), is charged as full minutes. Therefore, \(f(x) = 0.50 + 0.25 \times \lceil x - 1 \rceil\), where \(\lceil x - 1 \rceil\) rounds up to the nearest whole number. Thus, the piecewise function is: \[ f(x) = \begin{cases} 0.50, & 0< x \leq 1 \ 0.50 + 0.25 \times \lceil x - 1 \rceil, & 1 < x \leq 5 \end{cases} \]

Key Concepts

Function Notation

Function notation is a way to express mathematical relationships that makes it easy to understand how a function behaves. In our exercise, the function notation is used to express the cost of a phone call as a function of its duration in minutes. A function like this is written as \( f(x) \), where \( f \) is the function name, and \( x \) is the variable representing the input—in this case, the length of the call.

The purpose of function notation is to simplify the process of calculating outputs from given inputs. By substituting \( x \) with a specific number, you can quickly find the corresponding output value, such as the total cost of a phone call. This notation is particularly useful because it clearly defines the rules or formulas applied to the input to generate the output. For example, the notation helps us easily understand how the cost changes with the call duration. Function notation provides a structured way to model real-world situations mathematically.

In creating the function for call costs, we have a piecewise-defined function. This means different formulas apply depending on the value of \( x \). The function is defined as:

• \( f(x) = 0.50 \) for calls lasting 1 minute or less
• \( f(x) = 0.50 + 0.25 \times \lceil x - 1 \rceil \) for calls between 1 and 5 minutes

It empowers us to handle varying conditions within the same function efficiently.

Rounding Functions

Rounding functions are important when a calculation requires converting a decimal number to the nearest integer. In the context of this exercise, rounding plays a crucial role in determining the cost of phone calls that last for a non-integer number of minutes. Instead of charging for the exact fractional time, the rule is to round up. This is achieved with the ceiling function, \( \lceil x \rceil \), which rounds a number \( x \) up to the nearest integer.

For example, for a call lasting 3.5 minutes, we need to charge as though the call lasted for 4 full minutes after the first minute because we round up the additional 2.5 minutes to 3 full minutes. Understanding different types of rounding can help accurately calculate costs and make decisions that require an approximation. Here, the ceiling function ensures every fraction of a minute is treated as a full minute, to align with the rate structure provided.

Rounding up may seem like a small detail, but it can significantly affect cumulative calculations, especially in billing scenarios. Thus, comprehending rounding and specifically the application of ceiling functions is essential for precise financial and mathematical calculations.

Cost Calculation

Calculating costs involves understanding the pricing structure and applying it accurately in mathematical computations. In this problem, the cost calculation for a phone call is determined by a combination of fixed and variable costs. The fixed cost is \( \\(0.50 \) for the first minute, while each additional minute costs \( \\)0.25 \).

To determine the total cost, follow these steps:

• First minute: \( \\(0.50 \)
• Additional minutes: Count remaining minutes over the first one, rounded up to the next whole number.
• Cost of additional minutes: Multiply the number of additional minutes by \( \\)0.25 \)

For a 3.5-minute phone call, you will calculate costs as follows:

• The first minute costs \( \\(0.50 \)
• The remaining 2.5 minutes are rounded up to 3 minutes
• These 3 additional minutes cost \( 3 \times 0.25 = \\)0.75 \)
• Thus, total cost is \( \\(0.50 + \\)0.75 = \$1.25 \)

Understanding cost calculation helps in making informed financial choices and ensures accurate billing. It is crucial to grasp both the structure of the charges and the method of computation, as this knowledge is applicable in various real-world contexts involving fees and pricing.