Find the answers and explanations to exponential decay questions. Check your answers. Make sure you understand how to use an exponential function and problem solving skills to predict real world occurrences.

**Definitions**

**Exponential Decay:**An original amount is reduced by a consistent rate over a period of time.**Original Amount:**The amount before the decay occurs.**Decay Factor:**The percentage by which the original amount will decline. Students can study an exponential function, read a word problem, or compare data to find the decay factor.**Exponent**: Time, usually expressed in seconds, minutes, hours, days, or years.**Formula:***y*=*a(*1*-b)*: Final amount remaining after the decay over a period of time^{x}

y

*a*: The original amount

*b*: The decay factor

*x*: Time

### Exponential Decay In Real Life: The Decline of Newspaper Readership

Pennrose Gazette, the local newspaper for Pennrose County, is experiencing a reduction in circulation. These days, the Gazette's audience increasingly prefers the Internet for news.

In 2008, the Pennrose Gazette counted 20,000 readers. The publisher predicts that its readership will contract by 10% each year.

Use this information to answer the following questions.

### Answers and Explanations

Original Worksheet, *Exponential Decay Worksheet 2*

1. What was the original number of readers (*a*)?

**20,000 readers**

2. What is the decay factor (*b*)?

**10%**

3. Write a function describing the diminishing number of Pennrose Gazette readers.

Hint: *y* = *a(1 -b) ^{x}*

Remember to convert the decay factor, 10%, into a decimal to use in the function.

*y* = 20,000(1 - .10)* ^{x}* , where

*y*represents the remaining number of readers after

*x*

**years.**

4. Predictions: 2015

A. Write a function that can be used to predict Pennrose Gazette's readership in 2015.

*y* = 20,000(1 - .10)^{x}

Find *x*, the number of years that have passed since 2008.

*x* =2015 - 2008

*x* = 7

*y* = 20,000(1 - .10)^{7}

B. Based on the function, predict the number of readers in 2015.

*y* = 20,000(1 - .10)^{7}

*y =* 9,565.938

**In 2015, about 9,565 people will read the Pennrose Gazette.**

5.Predictions: 2028

A. Write a function that can be used to predict Pennrose Gazette's readership in 2028.

*y* = 20,000(1 - .10)^{x}

Find x, the number of years that have passed since 2008.

*x* =2028 - 2008

*x* = 20

*y* = 20,000(1 - .10)^{20}

B. Based on the function, predict the number of subscribers in 2028.

*y* = 20,000(1 - .10)^{20}

y = 2,431.533

**In 2028, about 2,431 people will read the Pennrose Gazette.**