Problem 42 - Applied Calculus

Question

Find the relative, or percent, change. \(R\) changes from 50 to 47

Step-by-Step Solution

Answer

The percent change is -6%.

Step 1: Identify the Initial and Final Values

To find the percent change, determine the initial value and the final value. In this exercise, the initial value \( R_{initial} \) is 50 and the final value \( R_{final} \) is 47.

Step 2: Calculate the Change in Value

Subtract the initial value from the final value to find the change in \( R \). The change is given by:\[\Delta R = R_{final} - R_{initial} = 47 - 50 = -3\]

Step 3: Calculate the Absolute Change Relative to the Initial Value

To find the relative change, divide the change by the initial value:\[\frac{\Delta R}{R_{initial}} = \frac{-3}{50} = -0.06\]

Step 4: Convert to Percent Change

To express the relative change as a percentage, multiply by 100:\[(-0.06) \times 100 \% = -6\%\] This represents a 6% decrease.

Key Concepts

Calculus Problem

In mathematics, calculus often involves understanding changes between variables. A common type of calculus problem is finding the rate of change, which is a concept that can be related to real-life contexts through elements like relative change or percent change. These problems are crucial for understanding how a quantity evolves over time.
The given exercise asks for the percent change of a number. This is a basic form of differential analysis that lets us know how a quantity changes from one state to another. While the exercise itself is simple, it mirrors more complex problems where changes are more nuanced and require calculus concepts like derivatives to solve.
In this context, percent change helps us quantify the rate of change between two points, which is essential in fields such as economics, physics, and data analysis.

Relative Change

Relative change, or percent change, provides a way to compare the magnitudes of changes relative to original values. It expresses the amount of change in terms of a percentage, giving a standardized way of evaluating differences.
For this exercise, we calculated the relative change by following these steps:

• First, identify the initial value (50) and the final value (47), which tells us about the shift in the value of a variable.
• Then, we find the difference (\(\Delta R\) or "change") between these values, which was -3 in this case.
• We express this change relative to the original amount by dividing by the initial value.
• Finally, to make the relative change more interpretable, we convert it into a percentage by multiplying by 100, resulting in a -6% change.

Negative values indicate a decrease, while positive values indicate an increase.

Mathematical Calculations

Mathematical calculations are integral in solving problems related to percent change. These calculations involve a series of operations, including subtraction, division, and multiplication. The goal is to accurately express the magnitude of changes in a way that provides meaningful information.
Here, the mathematical operations performed were simple yet powerful:

• Subtraction was used to determine the absolute change, showing how much the value decreased.
• A division was then employed to compare this change against the initial value, forming a ratio (relative change).
• Multiplication by 100 converted this ratio into a percentage, translating complex numerical information into an intuitive format that is easy to understand and discuss.

These computations not only answer the question but also enhance understanding of how variables relate through mathematical relationships. This process is a foundational skill in data analysis and various science disciplines.