Problem 41 - Applied Calculus

Question

Find the relative, or percent, change. \(W\) changes from 0.3 to 0.05

Step-by-Step Solution

Answer

The relative change is -83.33%.

Step 1: Identify Initial and Final Values

First, identify the initial value of \( W \) (which is 0.3) and the final value of \( W \) (which is 0.05). These values will help us find the percent change.

Step 2: Calculate the Change in Value

To find the change in value, subtract the initial value from the final value: \( 0.05 - 0.3 = -0.25 \). This change is negative, indicating a decrease.

Step 3: Compute the Relative Change

To calculate the relative change, divide the change in value by the initial value: \(-0.25 / 0.3 = -0.8333\).

Step 4: Convert to Percent Change

Convert the relative change to a percent change by multiplying by 100: \(-0.8333 \times 100 = -83.33\%\). This represents an 83.33% decrease.

Key Concepts

Relative Change

Relative change is a way to express the change in a quantity in relation to its starting value. It shows how much a certain value has increased or decreased compared to what it started out as. Relative change can be an increase or decrease:

• If the relative change is positive, it indicates an increase from the initial value.
• If negative, it means there's a decrease.

In the problem, the relative change is calculated as the difference between the final value and the initial value divided by the initial value. This gives us a way to see the change proportionally, rather than in absolute terms.

Initial Value

The initial value is the starting point in any measurement or calculation, and it's what changes are measured against. In this exercise, the initial value of the variable \(W\) is 0.3. Knowing and identifying the initial value is crucial in determining the relative change, because it acts as the reference from which all measurements are made. If you don't have the correct initial measurement, your percent or relative change will be inaccurate. Simply put, without the initial value:

• We cannot determine the direction (increase or decrease) of the change.
• We can't accurately quantify the change itself.

Final Value

The final value is the quantity or measurement you end up with after some process or change has occurred. In the given exercise, the final value of \(W\) is 0.05. Recognizing the final value is necessary for comparing against the initial value to see how much change has taken place. Similarly to the initial value, it provides a reference point:

• To assess whether the value has increased or decreased.
• To calculate the absolute and relative changes accurately.

In this context, understanding the final value is the key to determining if the observed change is indeed significant.

Calculus

Calculus, often regarded as an advanced mathematical field, provides tools to analyze changes. For percent change problems, especially involving rates of change, differential calculus can be exceptionally useful. Here's how calculus relates to percent change:

• Calculus provides tools for understanding how values change concerning one another via derivatives.
• The concept of a derivative helps in determining the rate at which one value changes relative to another, essentially making it a cornerstone of analyzing percent changes.

But in simpler terms, the percent change discussed here does not require calculus knowledge—it can be solved with straightforward arithmetic.