Problem 83 - Algebra and Trigonometry

Question

The domestic postage rate for first-class letters weighing 3.5 oz or less is 44 cents for the first ounce (or less), plus 17 cents for each additional ounce (or part of an ounce). Express the postage P as a piecewise-defined function of the weight \(x\) of a letter, with \(0< x \leq 3.5,\) and sketch the graph of this function.

Step-by-Step Solution

Answer

\[ P(x) = \begin{cases} 44, & 0 < x \leq 1, \\ 61, & 1 < x \leq 2, \\ 78, & 2 < x \leq 3, \\ 95, & 3 < x \leq 3.5 \end{cases} \]

Step 1: Understanding the Problem

We need to express the postage rate as a function of the weight of a letter. The rate starts at 44 cents for the first ounce and increases by 17 cents for each additional ounce or part of an ounce up to 3.5 ounces.

Step 2: Define the Postage Function

The function for the postage, denoted as \(P(x)\), changes based on the weight \(x\) of the letter. Each weight bracket represents an interval. We'll express the function using piecewise notation:

Step 3: Create Piecewise Function

Based on the rules, we define the function as:\[ P(x) = \begin{cases} 44, & 0 < x \leq 1, \61, & 1 < x \leq 2, \78, & 2 < x \leq 3, \95, & 3 < x \leq 3.5\end{cases} \]

Step 4: Graph the Function

To graph the function, plot the piecewise constant values on the cost \(P(x)\) vertically for each weight interval: - From \(0 < x \leq 1\), \(P(x) = 44\)- From \(1 < x \leq 2\), \(P(x) = 61\)- From \(2 < x \leq 3\), \(P(x) = 78\)- From \(3 < x \leq 3.5\), \(P(x) = 95\)

Step 5: Verify the Function's Definition

Ensure each interval correctly corresponds to a defined piece in the function. This confirms that as weight increases, the function accurately models the increase in postage costs according to the specified rules.

Key Concepts

Postage Rate

The concept of a postage rate is essential for understanding how mailing costs are calculated, particularly for sending first-class letters. A postage rate consists of the basic price you need to pay to send mail. In the given exercise, we are dealing with a specific domestic postage rate for first-class letters. For a letter weighing up to 3.5 ounces, the cost includes an initial charge for the first ounce, along with additional costs for any excess weight up to the limit. Understanding this initial rate and its increments for added ounces helps efficiently calculate mailing expenses.

The base rate for the first ounce is 44 cents. Each additional ounce or fraction thereof leads to an extra charge of 17 cents. The critical takeaway from this rate structure is its tiered nature: different weights incur different costs based on predefined intervals. This structured payment method makes mathematical representation through piecewise functions quite suitable.

First-Class Letters

First-class letters represent a category of mail that includes personal, private mailings such as letters and postcards. This is a convenient and economical way to communicate via mail. As the most common type of postal service for letters, first-class mail travels efficiently with a generally consistent delivery timeframe.

In this exercise, we focus on the pricing mechanics for letters designated as first-class. Understanding how first-class rates are calculated assists in financial planning, especially when sending multiple letters. These letters are prioritized, which means they also take precedence in the delivery queue, ensuring quicker arrival times compared to other non-priority mail services.

Weight

Weight is a key metric in determining postage costs because heavier items tend to require more resources for delivery. When dealing with first-class letters, the weight becomes a critical factor in how the postage rate is calculated.

In our case, the weights of the letters range up to 3.5 ounces. Breaking the weight down into segments helps establish intervals within which the postage cost changes. Each additional ounce or part results in a predetermined additional fee, emphasizing the importance of these intervals in postal pricing.

• 0 to 1 ounce: Base charge.
• 1 to 2 ounces: Adds the first additional charge.
• 2 to 3 ounces: Adds the second additional charge.
• 3 to 3.5 ounces: Adds the final additional charge.

Recognizing how weight impacts costs can help users better predict the necessary postage.

Graphing Functions

Graphing functions provides a visual representation that simplifies understanding the relationship between letter weight and its corresponding postage rate. In this exercise, a piecewise function is used to capture discrete changes in postage costs as the letter's weight increases.

To represent this piecewise function effectively:

• Plot the weight on the horizontal axis (x-axis).
• Plot the cost on the vertical axis (y-axis).

Each interval corresponds to a constant postal rate, illustrated by a horizontal line segment on the graph. For instance:

• From 0 to 1 ounce, the graph shows a straight line at 44 on the y-axis.
• From 1 ounce to just over 2 ounces, the line jumps to 61.

The graph visualizes how costs rise as weight increases, using separate flat segments to clearly denote changes at each weight bracket. This clarity aids in both understanding and communication, enabling a straightforward interpretation of how the postal rate function behaves across different weights.