Identifying Growth or Decay

Oct 13, 2025 by windailys.com 111 views

Introduction

Exponential functions are a fundamental concept in mathematics, describing how quantities change over time. In this article, we will explore the given exponential function, $y = 56(0.91)^x$ , and determine whether it represents growth or decay. We will also calculate the percentage rate of increase or decrease.

What is Exponential Growth and Decay?

Exponential growth and decay are two types of exponential functions that describe how quantities change over time. Exponential growth occurs when a quantity increases at a rate proportional to its current value, resulting in a rapid increase in the quantity. On the other hand, exponential decay occurs when a quantity decreases at a rate proportional to its current value, resulting in a rapid decrease in the quantity.

Identifying Growth or Decay

To determine whether the given exponential function represents growth or decay, we need to examine the base of the exponent. In the given function, $y = 56(0.91)^x$ , the base of the exponent is 0.91. Since the base is less than 1, it indicates that the function represents exponential decay.

Calculating the Percentage Rate of Increase or Decrease

To calculate the percentage rate of increase or decrease, we need to find the rate of change of the function. The rate of change can be calculated using the formula:

\text{Rate of Change} = \frac{\text{Change in Value}}{\text{Original Value}}

In this case, the original value is 56, and the change in value is the difference between the current value and the original value. To find the current value, we can use the given function:

y = 56(0.91)^x

Since we are not given a specific value of x, we will assume that x is a variable. To find the rate of change, we can take the derivative of the function with respect to x:

\frac{dy}{dx} = 56(0.91)^x \ln(0.91)

The rate of change is the derivative of the function, which is 56(0.91)^x ln(0.91). To find the percentage rate of increase or decrease, we can divide the rate of change by the original value and multiply by 100:

\text{Percentage Rate of Change} = \frac{56(0.91)^x \ln(0.91)}{56} \times 100

Simplifying the expression, we get:

\text{Percentage Rate of Change} = (0.91)^x \ln(0.91) \times 100

Since the base of the exponent is 0.91, which is less than 1, it indicates that the function represents exponential decay. Therefore, the percentage rate of decrease is:

\text{Percentage Rate of Decrease} = (0.91)^x \ln(0.91) \times 100

Example

Let's assume that x = 5. To find the current value, we can plug in x = 5 into the given function:

y = 56(0.91)^5

Using a calculator, we get:

y = 56(0.66)

y = 36.96

The current value is 36.96. To find the percentage rate of decrease, we can use the formula:

\text{Percentage Rate of Decrease} = \frac{56 - 36.96}{56} \times 100

Simplifying the expression, we get:

\text{Percentage Rate of Decrease} = \frac{19.04}{56} \times 100

\text{Percentage Rate of Decrease} = 34.07\%

Therefore, the percentage rate of decrease is approximately 34.07%.

Conclusion

In conclusion, the given exponential function, $y = 56(0.91)^x$ , represents exponential decay. The percentage rate of decrease can be calculated using the formula:

\text{Percentage Rate of Decrease} = (0.91)^x \ln(0.91) \times 100

Introduction

Exponential growth and decay are fundamental concepts in mathematics that describe how quantities change over time. In our previous article, we explored the given exponential function, $y = 56(0.91)^x$ , and determined whether it represents growth or decay. We also calculated the percentage rate of increase or decrease.

In this article, we will answer some frequently asked questions about exponential growth and decay. Whether you're a student, a teacher, or simply someone interested in mathematics, this Q&A guide will help you understand these complex concepts.

Q: What is the difference between exponential growth and decay?

A: Exponential growth occurs when a quantity increases at a rate proportional to its current value, resulting in a rapid increase in the quantity. On the other hand, exponential decay occurs when a quantity decreases at a rate proportional to its current value, resulting in a rapid decrease in the quantity.

Q: How do I determine whether an exponential function represents growth or decay?

A: To determine whether an exponential function represents growth or decay, you need to examine the base of the exponent. If the base is greater than 1, the function represents exponential growth. If the base is less than 1, the function represents exponential decay.

Q: How do I calculate the percentage rate of increase or decrease?

A: To calculate the percentage rate of increase or decrease, you need to find the rate of change of the function. The rate of change can be calculated using the formula:

\text{Rate of Change} = \frac{\text{Change in Value}}{\text{Original Value}}

In this case, the original value is the initial value of the function, and the change in value is the difference between the current value and the original value.

Q: What is the formula for calculating the percentage rate of increase or decrease?

A: The formula for calculating the percentage rate of increase or decrease is:

\text{Percentage Rate of Change} = \frac{\text{Rate of Change}}{\text{Original Value}} \times 100

Q: How do I use the formula to calculate the percentage rate of increase or decrease?

A: To use the formula, you need to plug in the values of the rate of change and the original value. For example, if the rate of change is 0.05 and the original value is 100, the percentage rate of increase or decrease would be:

\text{Percentage Rate of Change} = \frac{0.05}{100} \times 100

\text{Percentage Rate of Change} = 5\%

Q: What is the significance of the base of the exponent in an exponential function?

A: The base of the exponent in an exponential function determines whether the function represents growth or decay. If the base is greater than 1, the function represents exponential growth. If the base is less than 1, the function represents exponential decay.

Q: Can you give an example of an exponential function that represents growth?

A: Yes, an example of an exponential function that represents growth is:

y = 2(1.1)^x

In this function, the base of the exponent is 1.1, which is greater than 1. Therefore, the function represents exponential growth.

Q: Can you give an example of an exponential function that represents decay?

A: Yes, an example of an exponential function that represents decay is:

y = 100(0.9)^x

In this function, the base of the exponent is 0.9, which is less than 1. Therefore, the function represents exponential decay.

Conclusion

In conclusion, exponential growth and decay are fundamental concepts in mathematics that describe how quantities change over time. By understanding these concepts, you can calculate the percentage rate of increase or decrease and determine whether an exponential function represents growth or decay. We hope this Q&A guide has helped you understand these complex concepts.