10-6 Growth and Decay

Objective:Students will be able to solve problems

involving exponential growth or exponential decay.

Exponential Growth

• Exponential growth is when some initial amount increases by the same percent over a given period of time.

• It represents rapid growth.• It is very common to see in real life, especially

in science, as many microorganisms and bacteria grow exponentially.

Formula for Exponential Growth

• y represents the final amount• C represents the initial amount• r represents the rate of change (expressed as a

decimal)• t represents the time

Example 1: In 1971, there were 294,105 females participating in high school sports. Since then, that number has increased an average of 8.5% per year.Write an equation to represent the number of females participating in high school sports since 1971. Then find how many females participated in high school sports in the year 2001.

Example 2: In 2000, the US had a population of about 280 million, and a growth rate of about 0.85% per year. Write an equation to represent the population of the United States since the year 2000. Then find what the population will be in the year 2010.

Try this one. Example 3: The population of Harveytown in 1995 was 25,000. Since then, the population has grown at an average rate of 3.2% each year. Write an equation to represent the population of Harveytown since 1995. Then, find the population of Harveytown in the year 2005.

Compound Interest

• One special application of exponential growth is compound interest. Here is the formula:

• A represents the amount of the investment• P represents the principal (initial amount)• n represents the number of times that the interest

in compounded each year• t represents the number of years that the money

is invested

Example 4: When Walter Burgundy was born, his father, Ron Burgundy, invested $1000 in a fixed rate savings account at a rate of 7% compounded annually. The money will go to Walter when he turns 18 to help him with college expenses. What amount of money will Walter receive from the investment?

Try this. Example 5: Mr. Frew received a check for $12,000 for working as Buzz Lightyear’s stunt double. He decided to put it in a savings account that paid 3.5% interest, compounded monthly (hint: there are 12 months in a year). Find the balance of Mr. Frew’s account in 3 years.

Exponential Decay

• Exponential decay is when some initial amount decreases by the same percent over a period of time.

• The variables are the same as with exponential growth, however the formula is slightly different.

Example 6: During an economic recession, a charitable organization found that its donations dropped by 1.1% per year. Before the recession, its donations were $390,000. Write an equation to represent the charity’s donations since the beginning of the recession. Then, estimate the amount of the donations 5 years after the start of the recession.

Example 7: Mr. Rasczyk needs some new equipment for his side job as a lumberjack. He buys a new mechanical ax for $50,000. If the ax depreciates 10% per year, find the value of the ax in 7 years.

Try this. Example 8: Ms. Okul’s cat Jack and Ms. Harvey’s cat Pam decided to pool the money they have recently earned as extras acting in the new “Meow Mix” commercial in order to start up a new record label. They purchased the record label for $45,000. However, their first CD “Meow That’s What I Call Music: Volume 1” was a flop, and the value of their label decreased at a rate of 12% per year. Write an equation to represent the value of the label since it was purchased. Then find the value of the label in 5 years.

Jack Selfie!