Problem 57 - A Graphical Approach to College Algebra

Question

Cellular Phone Bills 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

Step-by-Step Solution

Answer

A call lasting 3.5 minutes costs $1.25. The function is piecewise with costs increasing by $0.25 every minute after the first.

Step 1: Understanding the Charge Structure

The initial charge for a phone call is \\(0.50 for the first minute. Each additional minute costs \\)0.25. If the call lasts longer than a whole number of minutes, you must round up the time to the nearest whole number.

Step 2: Calculate Total Minutes Used for 3.5 Minutes Call

Since calls are rounded up to the next whole minute, a call lasting 3.5 minutes is rounded up to 4 minutes. The effective duration for billing purposes is 4 minutes.

Step 3: Calculate the Cost for 3.5 Minutes Call

The cost of the first minute is \\(0.50. The cost for the remaining 3 minutes (since the total is 4 minutes) is \(3 \times \\)0.25 = \\(0.75\). So, the total cost is \(\\)0.50 + \\(0.75 = \\)1.25\).

Step 4: Define the Function for Call Costs

A piecewise function is defined based on the duration, \(x\): - If \(0 < x \leq 1\), then \(f(x) = \\(0.50\). - If \(1 < x \leq 2\), then \(f(x) = \\)0.75\). - If \(2 < x \leq 3\), then \(f(x) = \\(1.00\). - If \(3 < x \leq 4\), then \(f(x) = \\)1.25\). - If \(4 < x \leq 5\), then \(f(x) = \$1.50\).

Step 5: Generalize the Piecewise Function

The function can be expressed as:\[f(x) = \begin{cases} 0.50, & 0 < x \leq 1 \ 0.75, & 1 < x \leq 2 \ 1.00, & 2 < x \leq 3 \ 1.25, & 3 < x \leq 4 \ 1.50, & 4 < x \leq 5\end{cases}\]

Key Concepts

Cost Calculation

Calculating the cost of a service like an international phone call requires understanding the pricing structure involved. Phone companies often have specific rates for initial and subsequent time periods. In this example,

• the first minute of a call costs \(\\(0.50\), and
• each additional minute costs \(\\)0.25\).

This means if a call lasts more than one minute, the cost will increase by a fixed amount for each additional minute.

To calculate the cost of a call, simply apply the rate for the first minute, then multiply the remaining time by the rate for additional minutes. If a call lasts 3.5 minutes, round up to 4 minutes (explained in the rounding section). The first minute costs \(\\(0.50\), and the remaining 3 minutes each cost \(\\)0.25\), leading to a total cost of \(\$1.25\).

Understanding this structure helps in anticipating how costs rise with longer call durations.

Rounding Up

When calculating costs, especially for services charged by time, rounding is a pivotal concept. Here, we're rounding up any fraction of a minute to the nearest whole number. This rounding method is standard in billing processes to ensure clear and consistent charges.

In our phone call example, a call lasting 3.5 minutes would be rounded up to 4 minutes. Here’s why it matters:

• It simplifies billing by eliminating fractions, making charges more straightforward.
• It ensures that the bill reflects the possibility of service usage throughout the entire rounded period.

Always remember, in cost calculations, if any part of a minute is used, the whole minute will often be counted. This rounding practice can be crucial for businesses when applying charges fairly.

Mathematical Function

Piecewise functions are incredibly useful for situations where costs or other values change at specific intervals. In this phone billing example, a piecewise function perfectly models how the cost changes with different call lengths.

A piecewise function comprises multiple sub-functions, each applying to different parts of the domain. For phone call costs from 0 to 5 minutes:

• If 0 < \(x \leq 1\), cost is \(\\(0.50\).
• If 1 < \(x \leq 2\), cost is \(\\)0.75\).
• If 2 < \(x \leq 3\), cost is \(\\(1.00\).
• If 3 < \(x \leq 4\), cost is \(\\)1.25\).
• If 4 < \(x \leq 5\), cost is \(\$1.50\).

This structured approach allows you to determine the call cost for any duration within this range. Piecewise functions ensure calculations are clear and systematic; they can easily be adjusted for different rate intervals or changes in pricing.