Problem 90 - Algebra and Trigonometry

Question

Postage Rates The domestic postage rate for first-class letters weighing 12 oz or less is 37 cents for the first ounce (or less), plus 23 cents for each additional ounce (or part of an ounce). Express the postage \(P\) as a function of the weight \(x\) of a letter, with \(0 < x \leq 12,\) and sketch the graph of this function.

Step-by-Step Solution

Answer

Postage function: \( P(x) = 37 + 23 (\lceil x - 1 \rceil) \). Graph is a step function.

Step 1: Understand the Problem

Our task is to express the postage rate \( P \) as a function of the weight \( x \) for a letter weighing between 0 and 12 ounces, inclusive. Initially, the first ounce costs 37 cents, and for each additional full or partial ounce, the postage costs an additional 23 cents.

Step 2: Define the Function

We need to determine how the cost changes as the weight \( x \) varies. For an initial weight up to 1 ounce, the cost is 37 cents. For weights exceeding 1 ounce, each additional full or part of an ounce costs an additional 23 cents.

Step 3: Determine Costs for Additional Weight

For weights greater than 1 ounce up to 12 ounces, the additional weight can be calculated as \( \lceil x-1 \rceil \) ounces. Therefore, each extra ounce above the first incurs a 23-cent fee.

Step 4: Create the Postage Function

The postage function \( P(x) \) can then be expressed as:\[ P(x) = 37 + 23 ( \lceil x - 1 \rceil ) \]where \( \lceil \, \rceil \) denotes the ceiling function, which rounds up to the nearest whole number.

Step 5: Graph the Function

The graph of this function is a step function, starting at 37 cents for \( 0 < x \leq 1 \), then increasing by 23 cents at each integer ounce greater than 1, up to and including 12 ounces. Use plot points like \( P(1) = 37 \), \( P(2) = 60 \), continuing up to \( P(12) = 290 \).

Key Concepts

Ceiling Function

The ceiling function is an important concept in mathematics, especially when dealing with real numbers that need to be approximated to their next largest integer.
In simple terms, the ceiling function takes any real number and rounds it up to the nearest whole number, regardless of whether the number is already whole or not.
For example, the ceiling of 1.2 is 2, and the ceiling of 3.0 is 3 because it is already an integer.
This function is helpful in various scenarios where rounding up values is necessary. In the context of postage rates, the ceiling function calculates the cost for partial weights as full ounces.
This ensures that any letter weighing more than 1.0 ounces but less than 2.0 ounces is treated as a full 2 ounces, and so on.

• Ceiling of 1.5 is 2.
• Ceiling of 2.1 is 3.

Applying the ceiling function in our postage rate function allows us to charge accurately based on weight exceeding the first ounce.

Step Function

A step function is a type of piecewise function that changes values at specific intervals and remains constant within each interval.
It resembles a staircase when graphed, hence the name "step" function.
In our postage rate example, the function creating steps at each increase in weight illustrates how postage prices increase by 23 cents as the weight approaches the next whole number of ounces.
The step function here is described by the ceiling function, as each increase in weight beyond the first ounce results in a higher cost:

• Postage remains 37 cents for any weight between 0.1 and 1 ounce.
• At slightly above 1 ounce (including 1.1, 1.5, etc.), the rate jumps to 60 cents, reflecting the next step.

As a visual representation of pricing based on weights, step functions are valuable in making calculations predictable and systematic.

Postage Rate Calculation

Calculating postage rates involves understanding how incremental costs are applied to weight, particularly when using a function like: \[ P(x) = 37 + 23(\lceil x - 1 \rceil) \]Each additional ounce (or fraction of an ounce) beyond the first causes an increase in postage cost by 23 cents.
For example, if a letter weighs 3.2 ounces:

• The postage includes 37 cents for the first ounce.
• Then 23 cents for each of the next two full ounces (including the partial ounce, due to ceiling function rounding).

This calculation ensures transparent and fair pricing based on weight. By setting artbitrary ranges for weight in ounces and calculating costs with full or partial ounces considered as a whole, the rate calculation becomes fair and logical.

Function Graphing

Graphing piecewise functions like the step function used in postage rate calculation provides a clear visual representation of how costs change with weight.
The graph will appear like a series of flat, horizontal lines ("steps") that jump to higher values at integer weights.
This visual helps students and users to easily predict the cost for different weights. The graph for our postage function, starting at 37 cents, looks like a step going up by 23 cents for every integer increment in weight:

• The first step starts from weight just above 0 to 1, priced at 37 cents.
• The next step covers just above 1 ounce to 2 ounces, priced at 60 cents, and so on up to 12 ounces.

By locating any specific weight on the x-axis, one can quickly identify the corresponding postage rate on the y-axis. This makes function graphing an incredibly effective tool for visual learning and application in real-world scenarios.