Problem 48 - Algebra 2

Question

For each annual rate of change, find the corresponding growth or decay factor. $$ -75 \% $$

Step-by-Step Solution

Answer

The decay factor corresponding to -75% annual rate of change is 0.25.

Step 1: Identify the rate of change

We are given a rate of change of -75%. This indicates a decay because the percentage is negative.

Step 2: Convert the percentage to a decimal

The rate of change -75% is equivalent to -0.75 in decimal form. This is done by dividing the percentage by 100.

Step 3: Find the decay factor

From step 2, we know that the rate of decay is -0.75. To find the decay factor, we need to add this to 1 because this is a decrease. Therefore, the decay factor is \(1 - 0.75 = 0.25\).

Key Concepts

Rate of Change

Understanding the **rate of change** is crucial when dealing with scenarios where values increase or decrease over time. In our example, the rate of change is -75%. This negative sign suggests that we are observing a reduction, which is commonly referred to as exponential decay.

Exponential decay implies that something is decreasing at a continuous rate over intervals. The rate of change helps us quantify how fast or slow an amount diminishes.

• A positive rate signals growth, while a negative rate indicates decay.
• Email marketing unsubscribes, depreciation of car values, or diminishing investments due to withdrawals all could depict situations with negative rates.

By comprehending the magnitude illustrated by the rate, we can better predict future values based on current trends.

Decay Factor

A **decay factor** is key to understanding how exponential decay affects the value of an item or phenomenon over time. Essentially, it determines the remaining proportion after one period.

To calculate the decay factor from a negative rate of change, we subtract the decimal equivalent of the percentage from 1. Using our example:

• Given a rate of decay of -75%, when converted to decimal, we have -0.75.
• The decay factor is then calculated as: \(1 - 0.75 = 0.25\). This implies that 25% of the initial value remains after one period of decay.

Understanding the decay factor allows you to predict long-term effects by applying this factor over multiple intervals, such as years or months. For instance, if an investment loses 75% of its value each year, only 25% of the initial investment remains each year.

Percentage to Decimal Conversion

The conversion from **percentage to decimal** is a fundamental step in calculating both growth and decay factors. Simply, this conversion makes it easier to manipulate and use the rate of change in calculations.

To convert a percentage like -75% to a decimal, follow these steps:

• Remove the percentage sign which gives us 75.
• Recognize the negative indicates a decrease.
• Divide by 100 to convert the percentage to decimal form: 75 ÷ 100 = 0.75.

Additionally, being familiar with this conversion process helps in a wide array of scenarios, whether calculating discounts or assessing markups in financial contexts. The simplicity of moving from percentage to decimal makes the concept of decay factors more intuitive and manageable in mathematical and real-world applications.