Problem 48 - Applied Calculus

Question

On May 11,2009, the cost to mail a letter in the US was raised from 42 cents to 44 cents. Find the relative change in the cost.

Step-by-Step Solution

Answer

The relative change in cost is approximately 4.76%.

Step 1: Identify the Initial and New Values

The initial cost to mail a letter was 42 cents, and the new cost is 44 cents.

Step 2: Calculate the Change in Cost

Subtract the initial cost from the new cost: \[ \text{Change in Cost} = 44 \text{ cents} - 42 \text{ cents} = 2 \text{ cents} \]

Step 3: Determine the Relative Change Formula

Relative change can be calculated using the formula: \[ \text{Relative Change} = \frac{\text{Change in Cost}}{\text{Initial Cost}} \]

Step 4: Compute the Relative Change

Substitute the values into the formula: \[ \text{Relative Change} = \frac{2}{42} \approx 0.0476 \]

Step 5: Convert to Percentage

To express the relative change as a percentage, multiply by 100: \[ 0.0476 \times 100 \approx 4.76\% \]

Key Concepts

Understanding Percentage Increase

A percentage increase is a way to express how much a value has grown as a proportion of its original size. When something becomes more costly, such as mailing a letter, it means that the cost has experienced a percentage increase. Calculating percentage increase involves a few simple steps:

• Determine the change in value. In this context, it means finding the difference between the new and the old costs.
• Use the relative change formula to calculate the change as a fraction of the original cost.
• Convert the relative change into a percentage by multiplying by 100.

Once you've done this, you gain insight into how significant or minor the increase really is. This method is essential when evaluating price changes, whether they're for everyday items or more significant investments.

Identifying Initial and New Values

Understanding the initial and new values is foundational to calculating relative changes. The initial value is what you start with—in our case, the original cost of mailing a letter, which was 42 cents. The new value is the updated or changed amount, meaning the 44 cents after the increase. Identifying these values is crucial to perform any relative change calculation accurately.

• Initial value: The starting point, which helps gauge the magnitude of the increase or decrease.
• New value: The adjusted or current state you compare the initial value to.

These values are like the bookends of your calculation process that allow you to understand the shift in costs effectively.

Breaking Down Cost Calculation

Cost calculation helps in understanding the monetary difference between two points in time or two states of an entity. In our example, it's necessary to determine the amount by which the mailing cost has changed. Here's how you break it down step-by-step:

• Find the difference between the new and initial values: Subtract the initial cost from the new cost.
• Understand this difference as the actual change in cost: Here, it's an increase of 2 cents (44 cents - 42 cents = 2 cents).
• Use this change to further determine its impact in relative terms, such as percentage change.

This simple arithmetic sets the stage for deeper analysis, like understanding financial trends or estimating budget requirements. By mastering cost calculation, you're better equipped to navigate everyday expenses wisely.