Problem 32 - Algebra 2

Question

In 2006, the cost to mail a first-class letter was 39¢ for any weight up to and including 1 ounce. Each additional ounce or part of an ounce added 24¢ to the cost. Make a graph showing the postal rates to mail any letter from 0 to 8 ounces.

Step-by-Step Solution

Answer

Rates increase by 24¢ per ounce after the first ounce; graph is a step function.

Step 1: Understand the Problem

The cost to mail a letter depends on its weight. For a letter weighing up to 1 ounce, the cost is 39¢. Each additional ounce or fraction thereof adds another 24¢ to the total cost.

Step 2: Define the Weight Categories

Identify the weight brackets from 0 to 8 ounces. These will be 0-1 ounce, 1-2 ounces, 2-3 ounces, ..., up to 7-8 ounces.

Step 3: Calculate Costs for Each Category

- For 0 to 1 ounce, the cost is 39¢. - For 1 to 2 ounces, add 24¢ to 39¢: 39 + 24 = 63¢. - For 2 to 3 ounces, add another 24¢: 63 + 24 = 87¢. - Continue adding 24¢ for each additional ounce: - 3 to 4 ounces: 87 + 24 = 111¢, - 4 to 5 ounces: 111 + 24 = 135¢, - 5 to 6 ounces: 135 + 24 = 159¢, - 6 to 7 ounces: 159 + 24 = 183¢, - 7 to 8 ounces: 183 + 24 = 207¢.

Step 4: Create a Graph

Plot the weight intervals on the x-axis (from 0 to 8 ounces). Plot the corresponding costs on the y-axis (using 39¢, 63¢, 87¢, 111¢, 135¢, 159¢, 183¢, and 207¢). Connect these points to see the increase as a step graph, which shows fixed costs for weight ranges and jumps in cost at each new ounce.

Key Concepts

Understanding Step Functions

Step functions are a unique type of graph that display data as a series of flat sections or "steps." They clearly represent fixed values over specific intervals, making them perfect for situations like postal rates, where costs change discretely at certain points. In our exercise about mailing costs, the postal rate remains constant within each weight bracket, then it jumps when you move to the next bracket. This creates a graph where each "step" represents a different weight category, such as 0-1 ounce, 1-2 ounces, and so on up to 8 ounces.

• The bottom axis (x-axis) of the graph displays the range of weights.
• The side axis (y-axis) shows the corresponding cost to mail a letter.

The graph will look like a staircase, with each horizontal segment of the step corresponding to a weight category with a flat cost, and each vertical jump representing an additional cost as you exceed the top weight of an interval. Thus, step functions help visualize data where a change occurs at distinct intervals, making them extremely useful for representing situations like postage rates or similar scenarios.

Defining Piecewise Functions

Piecewise functions are mathematical expressions that consist of multiple sub-functions, each applicable to different intervals of the input (like weight in our example). They allow us to describe different rules or formulas depending on the conditions. In the mailing cost scenario, each step in the piecewise function corresponds to a certain weight range with its own distinct cost function.

• The first piece covers weights from 0 to 1 ounce, where the cost is 39¢.
• The second piece applies to weights between 1 and 2 ounces with a cost of 63¢.

We continue defining pieces up to the weight of 8 ounces, each adding 24¢ to the cost as the weight increases. Piecewise functions help us handle scenarios that don't fit a single mathematically uniform equation but need various expressions to model different conditions or phases of behavior. These functions are useful for real-world situations like our mailing problem, where simple, separate rules govern distinct ranges.

Performing Cost Calculation

Cost calculation in our exercise involves determining the expense for mailing weights ranging from 0 to 8 ounces. It's quite straightforward with the right approach. For any letter weighing up to 1 ounce, the cost starts at 39¢, then increases by a fixed amount, 24¢, for each additional ounce or part of an ounce.

To calculate:

• For a letter up to 1 ounce: Cost = 39¢.
• For 1 to 2 ounces: Cost = 39¢ + 24¢ = 63¢.
• For 2 to 3 ounces: Keep adding 24¢ to the previous total, so 63¢ + 24¢ = 87¢.

Repeat this pattern by adding 24¢ for each weight bracket up to the specified 8 ounces. Each step calculation explains how specific weights correlate to a step increase in cost. This structured setup allows for a simple increment method to find the cost for different weights, embedding clarity into how postal costs are structured based on weight categories.