Problem 49 - Applied Calculus

Question

On January \(27,2013,\) the cost to mail a letter in the US \(^{46}\) was raised from 45 cents to 46 cents. Find the relative change in the cost.

Step-by-Step Solution

Answer

The relative change in cost is approximately 2.22%.

Step 1: Identify Initial and New Costs

The initial cost of mailing a letter was 45 cents, and the new cost is 46 cents. These values will be used to calculate the relative change.

Step 2: Calculate the Difference in Cost

Find the difference between the new cost and the initial cost. This is done by subtracting the initial cost from the new cost: \(46 - 45 = 1\) cent.

Step 3: Compute the Relative Change

The relative change is calculated by dividing the difference in cost by the initial cost. Use the formula: \( \frac{\text{New Cost} - \text{Initial Cost}}{\text{Initial Cost}} = \frac{1}{45} \approx 0.0222 \).

Step 4: Convert the Relative Change to a Percentage

To express the relative change as a percentage, multiply the decimal by 100: \(0.0222 \times 100 \approx 2.22\%\).

Key Concepts

Understanding Initial Cost

In this exercise, the initial cost refers to the starting point of comparison when determining how much the cost of mailing a letter has changed. The initial cost serves as the baseline or reference value. For instance, in our example, the initial cost of mailing a letter is given as 45 cents. This means that before any increase, the cost to send a letter was 45 cents. Understanding this initial starting point is crucial because it allows us to measure how much the cost has shifted.

• The initial cost provides a reference point for understanding changes.
• The initial cost in our example was 45 cents.

Remember, the initial cost is not just a starting figure, it's what we're comparing the new cost against to determine the extent of any change. It serves as the foundation for calculating increases or other adjustments in price.

Calculating Percentage Increase

Percentage increase is a way to express the change in a value as a proportion of the initial amount. It's a useful metric as it provides context to the change, transcending specific quantities, allowing us to understand the magnitude of the change in relative terms.To find the percentage increase:

• First, identify the change by subtracting the initial cost from the new cost.
• In this example, the cost increased from 45 to 46 cents, resulting in a 1 cent increase.
• Then, calculate the relative change by dividing the increase by the initial cost. This gives the decimal form of the change: \( \frac{1}{45} \approx 0.0222 \).
• Finally, convert this to a percentage by multiplying by 100, resulting in a 2.22% increase.

By focusing on percentage increases, we gain insights into how significant the change is compared to the original value. It enables quick comparisons across different scenarios, regardless of the starting values.

Step-by-Step Calculation

Breaking down the calculation into clear, simple steps helps demystify the process of finding the relative change. Each step builds on the previous one to guide you toward the final percentage increase.Here's how you perform this step-by-step calculation:

• **Step 1:** Identify both the initial and new costs. Here, it's 45 cents initially and 46 cents now.
• **Step 2:** Calculate the difference. Subtraction provides the change amount: \(46 - 45 = 1 \) cent.
• **Step 3:** Compute the relative change. Use the formula for relative change \( \frac{\text{Difference}}{\text{Initial Cost}} \). So, \( \frac{1}{45} \approx 0.0222 \).
• **Step 4:** Turn this decimal into a percentage for easier understanding: \(0.0222 \times 100 = 2.22\% \).

Following these steps ensures a systematic approach, enabling clear and accurate computation of percentage increases from any initial cost to a new cost.